# Complex Matrix

We denote the set of $m×n$ matrices with complex entries by $M_{m×n}(C)$. The rules of addition, scalar multiplication, matrix-vector product, matrix multiplication and transpose derived for real matrices also hold for complex matrices.

### Conjugate Transpose

Let $A=[a_{ij}]∈M_{m×n}(C)$. Then the conjugate of $A$ is
$A=[a_{ij}]$
and the *conjugate transpose* of $A$ is
$A_{∗}=A_{T}$

Let $A∈M_{m×n}(C)$. $A$ is called *Hermitian* if $A_{∗}=A$.

### Properties of complex matrices

$(A_{∗)_{∗}=}A$ $(A+B)_{∗}=A_{∗}+B_{∗}$ $(αA)_{∗}=αA_{∗}$ $(AB)_{∗}=B_{∗}A_{∗}$ $(Az)_{∗}=z_{∗}A_{∗}$