Hermitian matrix - Practice Questions & MCQ

Hermitian matrix

A square matrix $\mathrm{A}=\left[\mathrm{a}_{\mathrm{ij}}\right]_{\mathrm{n} \times \mathrm{n}}$ is said to be Hermitian matrix if $a_{i j}=\overline{a_{j i}} \forall \mathrm{i}, \mathrm{j}$,
i.e. $A=A^\theta, \quad\left[\right.$ where $A^\theta$ is conjugate transpose of matrix $\left.A\right]$

We know that when we take the transpose of a matrix, its diagonal elements remain the same, and while taking conjugate we just change the sign from +ve to -ve and -ve to +ve for the imaginary part of all elements, So to satisfy the condition A^?  = A diagonal elements must not change, ⇒ all diagonal element must be purely real, 

E.g. 

Let, $A=\left[\begin{array}{ccc}3 & 3-4 i & 5+2 i \\ 3+4 i & 5 & -2+i \\ 5-2 i & -2-i & 7\end{array}\right]$
Then,

$
\begin{gathered}
\mathrm{A}^{\prime}=\left[\begin{array}{ccc}
3 & 3+4 i & 5-2 i \\
3-4 i & 5 & -2-i \\
5+2 i & -2+i & 7
\end{array}\right] \\
\therefore \mathrm{A}^\theta=\overline{\left(\mathrm{A}^{\prime}\right)}=\left[\begin{array}{ccc}
3 & 3-4 i & 5+2 i \\
3+4 i & 5 & -2+i \\
5-2 i & -2-i & 7
\end{array}\right]
\end{gathered}
$

here, A is Hermitian matrix as $\mathrm{A}=\mathrm{A}^\theta$

Note : 

For any square matrix say A,  with complex number entries, 

$\mathrm{A}+\mathrm{A}^\theta$ is a Hermitian matrix

$
\left[\because\left(\mathrm{A}+\mathrm{A}^\theta\right)^\theta=\mathrm{A}^\theta+\left(\mathrm{A}^\theta\right)^\theta=\mathrm{A}^\theta+\mathrm{A}\right]
$