Complex Matrix

A Complex Matrix is a Matrix for which all entries are Complex Numbers

$$\begin{array}{r}Z\in {\mathbb{C}}^{m\times n}\end{array}$$

Unique Operations

In addition to all of the operations a regular matrix has, such as multiplication, addition, and so forth, the complex nature of the entries of complex matrices impart additional possibilities.

Conjugate

$$Z=(z_{i,j})⟹\bar{Z}=({\bar{z}}_{i,j})$$

Hermitian

Pretty much everywhere (formulae) that Transpose is used for real matrices, the Hermitian is used for complex matrices.

$$Z^H=(\bar{Z}{)}^T$$

Hermitian Properties

• $(A^H{)}^H=A$
• $(A+B{)}^H=A^H+B^H$
• $(cA{)}^H=\bar{c}{A}^H$
• $(AB{)}^H=B^H{A}^H$

Hermetian Inner Product

$${\vec{y}}^H\vec{x}$$

The length squared of a complex vector is

$${\vec{z}}^H\vec{z}$$

Hermetian Symmetric Matrix

$$Z\text{ is a }S⟺Z^H=Z$$

Hermetian Orthogonal Matrix

Hermetian + Orthogonal = Unitary

$$\begin{array}{rl}Z\text{ is a }Q& ⟺q_i^H{q}_j=\{\begin{array}{ll}0& i\ne j\\ 1& i=j\end{array}\\ & ⟺Z^{H}Z=I\end{array}$$