On Bézoutians, Van der Monde matrices, and the Lienard-Chipart stability criterion. (English) Zbl 0677.15011
First it is shown how congruence by Van der Monde matrices reduces the Bézoutian to diagonal form. Then for a polynomial \(q(z)=z^ n+q_{n- 1}z^{n-1}+...+q_ 0\) with \(q_+(z)=\sum q_{2j}z^ j,\quad q_- (z)=\sum q_{2j+1}z^ j\) the following remarkable result is proved: q(z) is a Hurwitz polynomial if and only if the Bézoutian \(B(q_+,q_-)\) is positive definite and all \(q_ i\) are positive.
Reviewer: M.Voicu
MSC:
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 15A63 | Quadratic and bilinear forms, inner products |
| 93D20 | Asymptotic stability in control theory |
Keywords:
Lienard-Chipart stability criterion; Van der Monde matrices; Bézoutian; diagonal form; Hurwitz polynomialReferences:
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