# User:IssaRice/Linear algebra/List of matrix products

From Machinelearning

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) | , where and are invertible matrices | Many of the other decompositions can be seen as restrictions of this case: e.g. restricting , or restricting , or restricting to be orthonormal (in which case inverse and transpose (in the real case) and adjoint (in complex case) are the inverse).
| ||

is diagonalizable; this is also called the eigendecomposition/spectral decomposition of | where is invertible and is diagonal | is a diagonal matrix | tao notes, p. 167, 168. | |

is similar to / Relationship of the descriptions of in one basis vs another | where is invertible | see Tao notes p. 120 | ||

Singular value decomposition | , where are unitary, is diagonal, and is the conjugate transpose of | is a diagonal matrix, where and are orthonormal bases | ||

Schur decomposition | where is unitary and is upper triangular | is upper triangular, where 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 via elementary row operations | ||||

where and are products of elementary matrices and has rank | see tao notes p. 138 |