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?
inertia of a non-symmetric matrix
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1If 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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Given any square matrix $A$, one can consider the quadratic form $x^TAx$, for which the index of inertia is already defined. But doing so amounts to looking only at the symmetric part of $A$, $(A+A^T)/2$. The antisymmetric part $(A-A^T)/2$ contributes $0$ to the expression $x^TAx$. The index of inertia of $x^TAx$ has to do with eigenvalues of $(A+A^T)/2$, not of $A$.
But I can give an example in which considering $x^TAx$ for non-symmetric $A$ is meaningful: namely, the comparison of eigenvalues to the norm of $A$. The property $x^TAx \ge c\,\|Ax\|\,\|x\|\quad \forall x\in\mathbb R^n$ distinguishes the class of matrices for which the angle between $x$ and $Ax$ never exceeds $\cos^{-1}c$.