User:IssaRice/Linear algebra/List of matrix products

Factorization theorems etc.

Change of coordinate decompositions

The following decompositions can all be seen as a "simple" matrix sandwiched by two change of coordinate matrices, or equivalently as a "simple" matrix found by looking at it from the "right" bases.

Product name	Matrix notation	Change of coordinate matrix	Linear transformation/basis notation	Notes
A generic decomposition (as far as I know, this doesn't have a name)	$A=PBQ^{-1}$, where $P$ and $Q$ are invertible matrices	$[T]_{\sigma }^{\sigma }=[I]_{\beta }^{\sigma }[T]_{\alpha }^{\beta }[I]_{\sigma }^{\alpha }=[I]_{\beta }^{\sigma }[T]_{\alpha }^{\beta }([I]_{\alpha }^{\sigma })^{-1}$		Many of the other decompositions can be seen as restrictions of this case: e.g. restricting $P=Q$, or restricting $\alpha =\beta$, or restricting $\alpha ,\beta$ to be orthonormal (in which case inverse and transpose (in the real case) and adjoint (in complex case) are the inverse).
$A$ is diagonalizable; this is also called the eigendecomposition/spectral decomposition of $A$	$A=QDQ^{-1}$ where $Q$ is invertible and $D$ is diagonal	$[T]_{\sigma }^{\sigma }=[I]_{\beta }^{\sigma }[T]_{\beta }^{\beta }[I]_{\sigma }^{\beta }=[I]_{\beta }^{\sigma }[T]_{\beta }^{\beta }([I]_{\beta }^{\sigma })^{-1}$	$[T]_{\beta }^{\beta }$ is a diagonal matrix	tao notes, p. 167, 168.
$A$ is similar to $B$ / Relationship of the descriptions of $T:V\to V$ in one basis vs another	$A=QBQ^{-1}$ where $Q$ is invertible	$[T]_{\beta '}^{\beta '}=[I_{V}]_{\beta }^{\beta '}[T]_{\beta }^{\beta }([I_{V}]_{\beta }^{\beta '})^{-1}$		see Tao notes p. 120
Singular value decomposition	$A=U\Sigma V^{*}$, where $U,V$ are unitary, $\Sigma$ is diagonal, and $V^{*}$ is the conjugate transpose of $V$	$[T]_{\sigma }^{\sigma }=[I]_{\beta }^{\sigma }[T]_{\alpha }^{\beta }[I]_{\sigma }^{\alpha }$	$[T]_{\alpha }^{\beta }$ is a diagonal matrix, where $\alpha$ and $\beta$ are orthonormal bases
Schur decomposition	$A=QUQ^{-1}$ where $Q$ is unitary and $U$ is upper triangular	$[T]_{\sigma }^{\sigma }=[I]_{\beta }^{\sigma }[T]_{\beta }^{\beta }[I]_{\sigma }^{\beta }=[I]_{\beta }^{\sigma }[T]_{\beta }^{\beta }([I]_{\beta }^{\sigma })^{-1}$	$[T]_{\beta }^{\beta }$ is upper triangular, where $\beta$ is an orthonormal basis

Elementary operation decompositions

The following decompositions involve elementary operations. Since a matrix is invertible iff it is a product of elementary matrices iff it is a change of coordinate matrix, you might think elementary operation decompositions should somehow correspond to the change of coordinate decompositions from above. This sounds reasonable, but I haven't figured out the connection yet.

Product name	Matrix notation	Change of coordinate matrix	Linear transformation/basis notation	Notes
Inverting $A$ via elementary row operations	$E_{m}E_{m-1}\cdots E_{2}E_{1}A=I$
	$A=B{\begin{pmatrix}I_{r}&0_{r\times (n-r)}\\0_{(m-r)\times r}&0_{(m-r)\times (n-r)}\end{pmatrix}}C$ where $B$ and $C$ are products of elementary matrices and $A$ has rank $r$			see tao notes p. 138

See also

• wikipedia:Matrix decomposition
• User:IssaRice/Linear algebra/Classification of operators