Matrix

Complex Matrix

We denote the set of $m\times n$ matrices with complex entries by $M_{m\times n}(\mathbb{C})$. 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 $A=[a_{ij}]\in M_{m\times n}(\mathbb{C})$. Then the conjugate of $A$ is $\overline{A}=[{\overline{a}}_{ij}]$ and the conjugate transpose of $A$ is $A^{\ast }={\overline{A}}^T$

Let $A\in M_{m\times n}(\mathbb{C})$. $A$ is called Hermitian if $A^{\ast }=A$.

Properties of complex matrices

$(A^{\ast {)}^{\ast }=}A$ $(A+B{)}^{\ast }=A^{\ast }+B^{\ast }$ $(\alpha A{)}^{\ast }=\overline{\alpha }{A}^{\ast }$ $(AB{)}^{\ast }=B^{\ast }{A}^{\ast }$ $(A\vec{z}{)}^{\ast }={\vec{z}}^{\ast }{A}^{\ast }$