# 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