# Diagonalization of a Hermitian matrix

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

• What do you mean by "opening determinant"? Mar 9, 2018 at 12:16
• As in evaluation of the determinant (the standard approach of finding eigenvalues/eigenvectors)
– AJ_
Mar 9, 2018 at 12:17
• Don't use images in this way, it makes it impossible for future users to find your question by searching. Instead, take the time to transcribe the problem here. Mar 9, 2018 at 12:18

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 $P = P_2 P_1$

• The part of getting the eigenvectors is still not doable in a given time frame.That is to say, it is still a long tedious process.
– AJ_
Mar 10, 2018 at 12:38