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Is there any definition for the inertia of a real quadratic non-symmetric matrix? If yes, how to compute it? One can surely refer to the symmetric part of the matrix and then proceed using the standard theory (e.g. Sylveter's law of inertia). But does this really makes sense?

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    Inertia counts the number of eigenvalues that are positive, zero, and negative. How exactly would you apply this concept to **complex** eigenvalues?2012-07-24
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    yes, this is not obvious. But you can associate a quadratic form to the non-symmetric matrix A, and then count the related positive, negative and zero eigenvalues.2012-07-24
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    If your matrix is of even order, you are aware that it is possible for it to have no real eigenvalues at all?2012-07-24
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    I was rather referring to the the symmetric part of the matrix.2012-07-24
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    If you associate a quadratic form to a non-symmetric matrix, this is the same quadratic form obtained from the symmetric part of the matrix, so there is no clue in considering a non-symmetric matrix to generate a quadratic form.2012-07-24

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