More Types of Matrices

Symmetric matrix

If $A=A^{T}$. Subset of square matrices. $$

Skew Symmetric matrix

If $A=-A^{T}$. Subset of square matrices. All elements in main diagonal are $0$.

Note

Any square matrix can be expressed as a sum of a symmetric matrix and a skew-symmetric matrix.

Complex conjucate of a matrix

Suppose $A=(\,a_{\text{ij}}\,)_{n\times n}$. Complex conjucate matrix of $A$ is: $$

$$A^{\text{*}} = \overline{A^{T}} = (\,\overline{a_{\text{ji}}}\,)_{n\times n}$$

Hermitian matrix

A square matrix $A$ is said to be a Hermitian matrix iff $A=\overline{A^{T}}$.

Skew Hermitian matrix

A square matrix $A$ is said to be a Hermitian matrix iff $A=-\overline{A^{T}}$.

Note

Any square matrix can be expressed as a sum of a hermitian matrix and a skew-hermitian matrix.