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 $A^H=A$.

Properties

• All eigenvalues are real
• Eigenvectors of distinct eigenvalues are orthogonal
• By spectral theorem (not covered in this module), there are exists $n$ orthogonal eigenvectors.

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 $x \neq \underline{0}$.