Matrix Factorization
Contents
Preliminaries
Def: A matrix $A \in M_n$ is normal if $AA^∗ = A^∗A$, that is, if $A$ commutes with its conjugate transpose.
Def: A complex matrix $A$ is unitary if $AA^∗ = I$ or $A^∗A = I$, and a real matrix $B$ is orthogonal if $BB^T = I$ or $B^TB = I$.
There is no so-called “orthonormal” matrix. There is just an orthogonal matrix whose rows or columns are orthonormal vectors.
Notice that $$ \begin{gathered} U^*U = I \Longleftrightarrow U^{\star}UU^{\star} = IU^{\star} = U^{\star} \ \Longleftrightarrow UU^{\star} = (U^{\star})^{-1}U^{\star} = I \end{gathered} $$ 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: $A^{\star}A=AA^{\star}=I$
- Symmetric: $A = A^{T}$
- Orthogonal: $A^TA=AA^T=I$