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Math Help - Are Similar Matrices Necessarily Congruent?

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    Are Similar Matrices Necessarily Congruent?

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

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    Now I understand the answer should be no, but who can give me a brief conclusion?
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    Quote Originally Posted by kevin_chn View Post
    Now I understand the answer should be no, but who can give me a brief conclusion?
    go by the definitions:

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

    P^{-1}AP = B

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

    P^TAP = B

    where P^T is the transpose of the matrix P

    for similar matrices to be congruent as well, therefore, we must have P^{-1} = P^T, which will not always be true. you can find a counter-example to show this

    see here for a counter-example
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