User:IssaRice/Linear algebra/Classification of operators - Revision history

https://machinelearning.subwiki.org/w/index.php?action=history&feed=atom&title=User%3AIssaRice%2FLinear_algebra%2FClassification_of_operators User:IssaRice/Linear algebra/Classification of operators - Revision history 2026-05-19T10:08:34Z Revision history for this page on the wiki MediaWiki 1.41.2 https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3488&oldid=prev IssaRice at 13:45, 28 December 2021 2021-12-28T13:45:08Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:45, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l7">Line 7:</td> <td colspan="2" class="diff-lineno">Line 7:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is diagonalizable || There exists a basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable (there exists a basis <math>\beta</math> of <math>V</math> with respect to which <math>[T]_\beta^\beta</math> is a diagonal matrix) || || || This basis is not unique because we can reorder the vectors and also scale eigenvectors by a non-zero number to obtain an eigenvector. But there are at most <math>\dim V</math> distinct eigenvalues so the diagonal matrix should be unique up to order? This result holds even if <math>V</math> is merely a vector space with any field of scalars. || If <math>T</math> is the identity map, then every non-zero vector <math>v \in V</math> is an eigenvector of <math>T</math> with eigenvalue <math>1</math> because <math>Tv = 1v</math>. Thus every basis <math>\beta = (v_1,\ldots,v_n)</math> diagonalizes <math>T</math>. The matrix of <math>T</math> with respect to <math>\beta</math> is the identity matrix.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is diagonalizable || There exists a basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable (there exists a basis <math>\beta</math> of <math>V</math> with respect to which <math>[T]_\beta^\beta</math> is a diagonal matrix) || || || This basis is not unique because we can reorder the vectors and also scale eigenvectors by a non-zero number to obtain an eigenvector. But there are at most <math>\dim V</math> distinct eigenvalues so the diagonal matrix should be unique up to order? This result holds even if <math>V</math> is merely a vector space with any field of scalars. || If <math>T</math> is the identity map, then every non-zero vector <math>v \in V</math> is an eigenvector of <math>T</math> with eigenvalue <math>1</math> because <math>Tv = 1v</math>. Thus every basis <math>\beta = (v_1,\ldots,v_n)</math> diagonalizes <math>T</math>. The matrix of <math>T</math> with respect to <math>\beta</math> is the identity matrix.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || || <math>TT^* = T^*T</math></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || || <math>TT^* = T^*T</math> <ins style="font-weight: bold; text-decoration: none;">|| A normal operator has the additional property that it can be written as <math>T = S + A</math>, where <math>S</math> is a self-adjoint operator and <math>A</math> is an anti-self-adjoint operator, and where <math>S</math> and <math>A</math> are simultaneously diagonalizable using a single orthonormal basis ||</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3487&oldid=prev IssaRice at 13:38, 28 December 2021 2021-12-28T13:38:22Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:38, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> <del style="font-weight: bold; text-decoration: none;">|| </del>|| || 90-degree rotation of the plane?</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> || || 90-degree rotation of the plane?</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3486&oldid=prev IssaRice at 13:38, 28 December 2021 2021-12-28T13:38:01Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:38, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> || <ins style="font-weight: bold; text-decoration: none;">|| || 90-degree rotation of the plane?</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3485&oldid=prev IssaRice at 13:35, 28 December 2021 2021-12-28T13:35:57Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:35, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (aka Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian ) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint (aka skew-Hermitian or anti-Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3484&oldid=prev IssaRice at 13:35, 28 December 2021 2021-12-28T13:35:42Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:35, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l9">Line 9:</td> <td colspan="2" class="diff-lineno">Line 9:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || || <math>TT^* = T^*T</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || || <math>TT^* = T^*T</math></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<del style="font-weight: bold; text-decoration: none;"><math>T</math> is </del>Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<ins style="font-weight: bold; text-decoration: none;">aka </ins>Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint <ins style="font-weight: bold; text-decoration: none;">(aka skew-Hermitian or anti-Hermitian ) </ins>|| There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka<del style="font-weight: bold; text-decoration: none;">, </del>unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale. || ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale. || ||</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3483&oldid=prev IssaRice at 13:34, 28 December 2021 2021-12-28T13:34:46Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:34, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l17">Line 17:</td> <td colspan="2" class="diff-lineno">Line 17:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale. || ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale. || ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">Acknowledgments: Thanks to Philip B. for feedback on this page.</ins></div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3482&oldid=prev IssaRice at 13:34, 28 December 2021 2021-12-28T13:34:23Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:34, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l11">Line 11:</td> <td colspan="2" class="diff-lineno">Line 11:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || || <math>T = T^*</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary <del style="font-weight: bold; text-decoration: none;">|| </del>|| || <math>T^* = -T</math> ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary || || <math>T^* = -T</math> ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka, unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka, unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || <math>TT^* = T^*T = TT^{-1} = I</math> || This only works when the field of scalars is the complex numbers</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3481&oldid=prev IssaRice at 13:33, 28 December 2021 2021-12-28T13:33:47Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:33, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3">Line 3:</td> <td colspan="2" class="diff-lineno">Line 3:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="sortable wikitable"</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{| class="sortable wikitable"</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>! Operator kind !! Description in terms of eigenvectors !! Description in terms of diagonalizability !! Geometric interpretation !! Notes !! Examples</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>! Operator kind !! Description in terms of eigenvectors !! Description in terms of diagonalizability !! Geometric interpretation <ins style="font-weight: bold; text-decoration: none;">!! Algebraic property </ins>!! Notes !! Examples</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is diagonalizable || There exists a basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable (there exists a basis <math>\beta</math> of <math>V</math> with respect to which <math>[T]_\beta^\beta</math> is a diagonal matrix) || || This basis is not unique because we can reorder the vectors and also scale eigenvectors by a non-zero number to obtain an eigenvector. But there are at most <math>\dim V</math> distinct eigenvalues so the diagonal matrix should be unique up to order? This result holds even if <math>V</math> is merely a vector space with any field of scalars. || If <math>T</math> is the identity map, then every non-zero vector <math>v \in V</math> is an eigenvector of <math>T</math> with eigenvalue <math>1</math> because <math>Tv = 1v</math>. Thus every basis <math>\beta = (v_1,\ldots,v_n)</math> diagonalizes <math>T</math>. The matrix of <math>T</math> with respect to <math>\beta</math> is the identity matrix.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is diagonalizable || There exists a basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable (there exists a basis <math>\beta</math> of <math>V</math> with respect to which <math>[T]_\beta^\beta</math> is a diagonal matrix) || <ins style="font-weight: bold; text-decoration: none;">|| </ins>|| This basis is not unique because we can reorder the vectors and also scale eigenvectors by a non-zero number to obtain an eigenvector. But there are at most <math>\dim V</math> distinct eigenvalues so the diagonal matrix should be unique up to order? This result holds even if <math>V</math> is merely a vector space with any field of scalars. || If <math>T</math> is the identity map, then every non-zero vector <math>v \in V</math> is an eigenvector of <math>T</math> with eigenvalue <math>1</math> because <math>Tv = 1v</math>. Thus every basis <math>\beta = (v_1,\ldots,v_n)</math> diagonalizes <math>T</math>. The matrix of <math>T</math> with respect to <math>\beta</math> is the identity matrix.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is normal || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> || <math>T</math> is diagonalizable using an orthonormal basis || <ins style="font-weight: bold; text-decoration: none;">|| <math>TT^* = T^*T</math></ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real <ins style="font-weight: bold; text-decoration: none;">|| || <math>T = T^*</math> </ins>||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary ||</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary <ins style="font-weight: bold; text-decoration: none;">|| || || <math>T^* = -T</math> </ins>||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka, unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry (aka, unitary in a complex vector space, or orthogonal in a real vector space) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || <ins style="font-weight: bold; text-decoration: none;">|| <math>TT^* = T^*T = TT^{-1} = I</math> </ins>|| This only works when the field of scalars is the complex numbers</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale. <ins style="font-weight: bold; text-decoration: none;">|| ||</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3480&oldid=prev IssaRice at 13:30, 28 December 2021 2021-12-28T13:30:07Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:30, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13">Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary ||</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary ||</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry <ins style="font-weight: bold; text-decoration: none;">(aka, unitary in a complex vector space, or orthogonal in a real vector space) </ins>|| There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || This only works when the field of scalars is the complex numbers</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is positive (positive semidefinite) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with nonnegative real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all nonnegative real numbers || Polar decomposition says an arbitrary linear operator can be written as a positive operator followed by a rotation (isometry). In polar decomposition, the positive operator step chooses orthogonal directions in which to stretch or shrink, so that we have a tilted ellipse, and the isometry rotates that ellipse. So a positive operator is simply one that does not require the second step. In other words, for a positive operator you can find some orthogonal "coordinate axes" along which to scale.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|}</div></td></tr> </table>

IssaRice https://machinelearning.subwiki.org/w/index.php?title=User:IssaRice/Linear_algebra/Classification_of_operators&diff=3479&oldid=prev IssaRice at 13:29, 28 December 2021 2021-12-28T13:29:33Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:29, 28 December 2021</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l10">Line 10:</td> <td colspan="2" class="diff-lineno">Line 10:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> self-adjoint (<math>T</math> is Hermitian) || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with real eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all real || </div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr> <tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| <math>T</math> is anti-self-adjoint || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> with pure imaginary eigenvalues || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries are all pure imaginary ||</ins></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || This only works when the field of scalars is the complex numbers</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| <math>T</math> is an isometry || There exists an orthonormal basis of <math>V</math> consisting of eigenvectors of <math>T</math> whose eigenvalues all have absolute value 1 || <math>T</math> is diagonalizable using an orthonormal basis and the diagonal entries all have absolute values 1 || || This only works when the field of scalars is the complex numbers</div></td></tr> </table>

IssaRice