Miscellaneous Facts about Matrices and Decomposition

Symmetric Matrix

If the matrix A is real, then a square matrix A is symmetric if $A=A^T$. However, for complex matrices, the matrix is “symmetric | Hermitian” $⟺A=A^H$ is a Hermitian Matrix (conjugate transpose of $\mathbf{A}$).

Characteristics of Real Symmetric Matrices:

• They have real eigenvalues.
•  A symmetric matrix can have negative eigenvalues.
• Their eigenvectors corresponding to distinct eigenvalues are orthogonal.
• They are always diagonalizable. (Spectral Theorem)

Positive Semi Definite Matrix

A symmetric or Hermitian matrix is PSD if for any vector x, ${\vec{x}}^{T}A\vec{x}\ge 0$.

Positive Semi Definite need not to be Symmetric and obviously, symmetric matrices need not to be Positive Semi Definite.

Positive Semi Definite Matrix ($A\ge 0$)

• $⟺$ all eigenvalues are non-negative.
• Cholesky Decomposition: $⟺$ there exists a matrix B such that $B^{T}B=A$. B is a lower triangular matrix.
• Singular values = Eigen values $\sigma =|\lambda |$

For general matrices (non-symmetric, non-Hermitian), there’s no direct equality between eigenvalues and singular values.