Hermitian

Refers to the operation of transposing a matrix and taking the complex conjugate for all elements. Denoted by $A^{H}$.

Hermitian Matrix

Matrix $A$ is a Hermitian matrix iff $A^H=A$.

For a Hermitian matrix $A$:

• All eigenvalues are real
• Eigenvectors of distinct eigenvalues are orthogonal
• By spectral theorem (not covered in this module), there are exists $n$ orthogonal eigenvectors.
• Eigenvectors of $A$ form a basis for $\mathbb{C}^{n\times 1}$

For any matrix $B_{m\times n}$:

• $BB^H$ and $B^HB$ are Hermitian matrices and have the same positive eigenvalues.
• $\text{rank}(B^HB) = \text{rank}(BB^H) = \text{rank}(B^H) = \text{rank}(B)$ is equal to the number of positive eigenvalues of $B^HB$.

Unitary Matrix

Matrix $P \in C_{n×n}$ Unitary iff: $P^HP= PP^H = I$ (i.e. $P^{-1} = P^H$).

If columns of $P$ are orthonormal then P is Unitary.

Positive Definite Matrix

Matrix $A\in C_{n×n}$ is Positive Definite iff $X^HAX>0$ for all $X \in \mathbb{C}^n - \set{\underline{0}}$.

For a Hermitian matrix $A$ with $\set{\lambda_1, \lambda_2,\dots,\lambda_n}$ eigenvalues:

Principal Minor

Principal minor of $A$ of order $k$ is the determinant of the matrix produced by deleting last $n-k$ rows and columns.

If all principle minors are positive (of order $n=1$ to $n$) when A is Positive Definite.