User:IssaRice/Linear algebra/List of matrix products

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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