Thread: Are Similar Matrices Necessarily Congruent?

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

Thank you~

Now I understand the answer should be no, but who can give me a brief conclusion?

Originally Posted by kevin_chn

Now I understand the answer should be no, but who can give me a brief conclusion?

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