Matrix Factorization

Preliminaries

Def: A matrix $A\in M_n$ is normal if $A{A}^{\ast }=A^{\ast }A$, that is, if $A$ commutes with its conjugate transpose.

Def: A complex matrix $A$ is unitary if $A{A}^{\ast }=I$ or $A^{\ast }A=I$, and a real matrix $B$ is orthogonal if $B{B}^T=I$ or $B^{T}B=I$.

There is no so-called “orthonormal” matrix. There is just an orthogonal matrix whose rows or columns are orthonormal vectors.

Notice that

$$\begin{array}{c}{U}^{\ast }U=I⟺U^{\star }U{U}^{\star }=I{U}^{\star }=U^{\star }\\ ⟺U{U}^{\star }=(U^{\star }{)}^{-1}{U}^{\star }=I\end{array}$$

the columns of $U$ are orthonormal if and only if the rows are orthonormal.

So the definition can be summarized as below:

• Hermitian: $A=A^{\star }$
• Unitary: $A^{\star }A=A{A}^{\star }=I$
• Symmetric: $A=A^T$
• Orthogonal: $A^{T}A=A{A}^T=I$

Eigenvalue Decomposition

Singular Value Decomposition

$LU$ Factorization

$QR$ Factorization