Answer to Question #215211 in Linear Algebra for Majhar

Question

Give an example, with justification, of a 

skew-Hermitian operator on C².

Answer 1

Let there be a matrix A

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

Now A is skew Hermitian only if

( A^* )^T = - A

Now we take the Conjugate of A and that is given as

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

On taking the transpose of above we have

( A^* )^T = $\begin{bmatrix} -i & -1+i & -2\\ 1+i &-3i & -i\\ 2&-i&0 \end{bmatrix}$

Now,

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

On comparing ( A^* )^T and - A we see that

( A^* )^T = - A

Hence, A is a skew Hermitian matrix.