## Franklin's Notes

### Matrix factorizations

This note is meant as a sort of list/index/cheatsheet of important matrix factorizations. Here are the ones that I've studied so far:

• Eigenvalue decomposition : $A=X\Lambda X^{-1}$ where $A\in \mathbb C^{m\times m}$ and $\Lambda\in \mathbb{C}^{m\times m}$ is diagonal

• Singular value decomposition : $A=U\Sigma V^\ast$ where $A\in\mathbb C^{m\times n}$ and $U\in\mathbb {C}^{m\times m},V\in\mathbb {C}^{n\times n}$ are unitary and $\Sigma\in\mathbb C^{m\times n}$ is diagonal

• QR factorization : $A=QR$ for a matrix $A\in\mathbb{C}^{m\times n}$, where $Q\in\mathbb{C}^{m\times m}$ is unitary and $R\in\mathbb{C}^{m\times n}$ is upper-triangular.

• LU factorization : $A=LU$ for a matrix $A\in\mathbb C^{m\times m}$, where $L\in \mathbb C^{m\times m}$ is lower triangular and $U\in \mathbb C^{m\times m}$ is upper triangular.

• Cholesky factorization : $A=L^\ast L$ for a Hermitian matrix $A\in\mathbb C^{m\times m}$ and a lower triangular matrix $L\in\mathbb C^{m\times m}$.

• Schur factorization : $A=QUQ^\ast$ for a matrix $A\in\mathbb C^{m\times m}$, where $Q\in\mathbb C^{m\times m}$ is unitary and $U\in\mathbb C^{m\times m}$ is upper triangular.

• Hessenberg reduction : $A=QHQ^\ast$ for a matrix $A\in\mathbb C^{m\times m}$, where $Q\in \mathbb C^{m\times m}$ is unitary and $H\in\mathbb C^{m\times m}$ is upper Hessenberg.