Diagonalization of a Hermitian matrix

Question

The problem I am facing is that with the standard approach of opening determinant it's taking a lot of time and is still not getting reduced.

Is there any other way to do this? Or any CLEANER way to reduce the determinant.

Thanks

Answer 1

If you do Gaussian elimination on an Hermitian matrix via a congruence transformation it also diagonalizes. With

$$ P_1 = \begin{pmatrix}1&0&0\\i&1&0\\-2&-1&1\end{pmatrix} $$

$\bar A=P_1AP_1^H$ gives

$$ \bar A = \begin{pmatrix}1&0&0\\ 0&1&-1 + i\\ 0&-1-i&2 \end{pmatrix} $$ Do it again on $\bar A$ and obtain another $P_2$

Then the required matrix is $P = P_2 P_1$