Matrices over Complex Numbers

Here the entries of the matrix are complex numbers. All the definitions still hold. One just needs to look at the following additional definitions.

DEFINITION 1.4.1 (Conjugate Transpose of a Matrix)  

1. Let $A$ be an $m \times n$ matrix over ${\mathbb{C}}.$ If $A=[a_{ij}]$ then the Conjugate of $A,$ denoted by $\overline{A},$ is the matrix $B= [b_{ij}]$ with $b_{ij} = \overline{a_{ij}}.$

   For example, Let $A=\begin{bmatrix}1 & 4 + 3 i & i \\ 0 & 1 & i - 2 \end{bmatrix}.$ Then

   $$\overline{A} =\begin{bmatrix}1 & 4 - 3 i & - i \\ 0 & 1 & -i - 2 \end{bmatrix}.$$

2. Let $A$ be an $m \times n$ matrix over ${\mathbb{C}}.$ If $A=[a_{ij}]$ then the Conjugate Transpose of $A,$ denoted by $A^*,$ is the matrix $B= [b_{ij}]$ with $b_{ij} = \overline{a_{ji}}.$

   For example, Let $A=\begin{bmatrix}1 & 4 + 3 i & i \\ 0 & 1 & i - 2 \end{bmatrix}.$ Then

   $$A^* =\begin{bmatrix}1 & 0 \\ 4 - 3 i & 1 \\ - i & -i - 2 \end{bmatrix}.$$

3. A square matrix $A$ over ${\mathbb{C}}$ is called Hermitian if $A^* = A.$
4. A square matrix $A$ over ${\mathbb{C}}$ is called skew-Hermitian if $A^{*} = -A.$
5. A square matrix $A$ over ${\mathbb{C}}$ is called unitary if $A^{*}A = A A^{*} = I.$
6. A square matrix $A$ over ${\mathbb{C}}$ is called Normal if $A A^{*} = A^{*} A.$

Remark 1.4.2   If $A=[a_{ij}]$ with $a_{ij} \in {\mathbb{R}},$ then $A^* = A^t.$

EXERCISE 1.4.3  

1. Give examples of Hermitian, skew-Hermitian and unitary matrices that have entries with non-zero imaginary parts.
2. Restate the results on transpose in terms of conjugate transpose.
3. Show that for any square matrix $A,$ $S = \frac{A+A^*}{2}$ is Hermitian, $T = \frac{A- A^*}{2}$ is skew-Hermitian, and $A = S + T.$
4. Show that if $A$ is a complex triangular matrix and $A A^* = A^* A$ then $A$ is a diagonal matrix.