# Hermitian and Skew - Hermitian Matrices

### Hermitian Matrix:

A square matrix $A$ is Hermitian if it is equal to its conjugate transpose: $A={A}^{†}$

• Properties:
• For a real matrix, being Hermitian is equivalent to being symmetric.
• For a complex matrix, the elements satisfy ${a}_{ij}=\stackrel{‾}{{a}_{ji}}$ for all $i$ and $j$.
• The diagonal elements must be real.
• Hermitian matrices are always square matrices.

### Example of Hermitian Matrix:

$A=\left[\begin{array}{cc}2& 5i\\ -5i& 3\end{array}\right]$

In this case, $A={A}^{†}$, which makes it a Hermitian matrix.

### Skew-Hermitian Matrix:

A square matrix $A$ is skew-Hermitian if it is equal to the negative of its conjugate transpose: $A=-{A}^{†}$

• Properties:
• For a real matrix, a skew-Hermitian matrix is equivalent to a skew-symmetric matrix.
• For a complex matrix, the elements satisfy ${a}_{ij}=-\stackrel{‾}{{a}_{ji}}$ for all $i$ and $j$.
• The diagonal elements must be purely imaginary.

### Example of Skew-Hermitian Matrix:

$A=\left[\begin{array}{cc}0& -7i\\ 7i& 0\end{array}\right]$

In this case, $A=-{A}^{†}$, which makes it a skew-Hermitian matrix.

### Properties of Hermitian and Skew-Hermitian Matrices:

1. Sum of Hermitian Matrices: The sum of two Hermitian matrices is also Hermitian. $A+B=C$ where $A$, $B$, and $C$ are Hermitian matrices.

2. Sum of Skew-Hermitian Matrices: The sum of two skew-Hermitian matrices is also skew-Hermitian. $A+B=C$ where $A$, $B$, and $C$ are skew-Hermitian matrices.

3. Product with Scalar: Both Hermitian and skew-Hermitian matrices preserve their property when multiplied by a scalar.