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enter image description here

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

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    $\begingroup$ What do you mean by "opening determinant"? $\endgroup$ Mar 9, 2018 at 12:16
  • $\begingroup$ As in evaluation of the determinant (the standard approach of finding eigenvalues/eigenvectors) $\endgroup$
    – AJ_
    Mar 9, 2018 at 12:17
  • $\begingroup$ 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. $\endgroup$ Mar 9, 2018 at 12:18

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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 $P = P_2 P_1$

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  • $\begingroup$ 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. $\endgroup$
    – AJ_
    Mar 10, 2018 at 12:38

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