Question

Which among the following is/are hermitian matrix

• A. $\left[\begin{array}{cc}1& i\\ -i& 1\end{array}\right]$
• B. $\left[\begin{array}{ccc}i& i& 5\\ i& -i& 0\\ -5& 0& 3i\end{array}\right]$
• C. $\left[\begin{array}{cc}i+1& 3i\\ 3i& i+1\end{array}\right]$
• D. $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Answer 1

Answer: (A), (D)

$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

We already saw that a Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate Transpose.

Here in option a : the transpose will be $\left[\begin{array}{cc}1& -i\\ i& 1\end{array}\right]$ . And the conjugate of this will be $\left[\begin{array}{cc}1& -i\\ i& 1\end{array}\right]$ which is same as original matrix

Here in option b : the transpose will be $\left[\begin{array}{ccc}i& i& -5\\ i& -i& 0\\ 5& 0& 3i\end{array}\right]$ And the conjugate of this will be $\left[\begin{array}{ccc}-i& -i& -5\\ -i& i& 0\\ 5& 0& -3i\end{array}\right]$ which is not the same as original matrix

Here in option c : the transpose will be $\left[\begin{array}{cc}i+1& 3i\\ 3i& i+1\end{array}\right]$. And the conjugate of this will be $\left[\begin{array}{cc}-i+1& -3i\\ -3i& -i+1\end{array}\right]$ which is not the same as original matrix

Here in option d : the transpose will be $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$. And the conjugate of this will be $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ which is same as original matrix