As two similar matrices have a same set of eigenvalues, are they necessarily congruent?

Thank you~

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- January 17th 2008, 01:02 AMkevin_chnAre Similar Matrices Necessarily Congruent?
As two similar matrices have a same set of eigenvalues, are they necessarily congruent?

Thank you~ - January 17th 2008, 01:24 AMkevin_chn
Now I understand the answer should be no, but who can give me a brief conclusion?

- January 17th 2008, 07:18 PMJhevon
go by the definitions:

an nxn matrix is similar to an nxn matrix if there exists an (invertible) nxn matrix such that:

an nxn matrix is congruent to an nxn matrix if there exists an (invertible) nxn matrix such that:

where is the transpose of the matrix

for similar matrices to be congruent as well, therefore, we must have , which will not always be true. you can find a counter-example to show this

see here for a counter-example