A generalization of the criterion that a quadratic form is positive definite (Q1910621)

This short note is concerned with the proof of the following Theorem: If for the real \(n \times n\)-matrix \(A\) a positive definite diagonal matrix \(K = \text{diag} (k_1, k_2, \dots, k_n)\) exists such that \(KA = A^TK\) then for the eigenvalues \(\lambda_i (A)\) of \(A\) \((i = 1, \dots, n)\) the following holds: (i) \(\lambda_i (A) > 0\) \(i = 1, \dots, n\) if and only if \(D_i (A) > 0\) for \(i = 1, \dots, n\). (ii) \(\lambda_i (A) < 0\) \(i = 1, \dots, n\) if and only if \((-1)^i D_i (A) > 0\) \(i = 1, \dots, n\). Here \(A^T\) denotes the transpose of \(A\) and \(D_i (A)\) are the leading principal minors of \(A\). This theorem generalizes a well known criterion for a quadratic form to be positive definite. The proof is elementary and is reduced to the above mentioned criterion.