Question

Show that the definition of a Hermitian matrix $\left(A=A^t\right)$can be writtenrole="math" localid="1658814044380" ${\mathit{a}}_{\mathbf{ij}}\mathbf{=}{\overline{\mathbf{a}}}_{\mathbf{ji}}$(that is, the diagonal elements are real and the other elements have the property that${\mathit{a}}_{\mathbf{12}}\mathbf{=}{\overline{\mathbf{a}}}_{\mathbf{21}}$, etc.). Construct an example of a Hermitian matrix.

Short answer

The Hermitian matrix is$A=A^T$

Step-by-step solution

Given information.

To show that the definition of a Hermitian matrix $\left(A=A^{*}\right)$can be written $a_{ij}=a_{ji}($that is, the diagonal elements are real and the other elements have the property that $a_{12}=a_{21}$, etc.). Construct an example of a Hermitian matrix.

Hermitian matrix

A Hermitian matrix is one in which the conjugate transpose of a complex square matrix is equal to itself.

Prove that the Hermitian matrix is A=AT.

By definition, $A=A^T$.

Hence the complex conjugate transpose of any matrix A is equal to A.

A contains the element $a_{ij}$.

Where,

represent rows and j represent columns.

Transpose means rows and columns getting interchanged.

Performing transpose operation,

localid="1658814788983" $\left(a_{ij}\right)=\left(a_{ji}\right)$

Its complex conjugate is,

$\left(a_{ij}\right)=\left({\overline{a}}_{ji}\right)$.

The example for Hermitian matrix is,

$$\begin{gathered} A=\left(11-i2 \\ 1+i3i \\ 2-i0\right) \end{gathered}$$

Hence,

$A^T=A$

$A^T=A$