关于赫维茨型矩阵多项式的系数
On the Coefficients of Hurwitz-Type Matrix Polynomials
浏览论文内容
中文总结 AI 辅助
该研究推导了赫维茨型矩阵多项式系数的显式公式，引入块赫维茨矩阵并建立相关行列式恒等式，还证明其系数矩阵行列式正定性在次数≤3时成立、次数为4时不成立，解决了相关猜想。
中文摘要 AI 辅助
考虑矩阵多项式 $\boldsymbol f_n(z)=I_qz^n+A_1z^{\n-1}+\boldsymbol{\text{⋯}}+A_n$，其中 $A_j \boldsymbol{\text{∈}} \boldsymbol{\text{ℂ}}^{q \times q}$。将其写为 $\boldsymbol f_n(z)=\boldsymbol h_n(z^2)+z\boldsymbol g_n(z^2)$。若矩阵多项式 $\boldsymbol f_n$ 满足：当 $n=2m$ 时，$\boldsymbol g_n(z)\boldsymbol h_n(z)^{-1}$；当 $n=2m+1$ 时，$\boldsymbol h_n(z)\bigl(z\boldsymbol g_n(z)\bigr)^{-1}$ 具有正定系数的有限连分式展开，则称其为赫维茨型矩阵多项式。我们推导了赫维茨型矩阵多项式系数的显式公式，涉及正交矩阵多项式、马尔可夫参数和舒尔补；引入了相关的块赫维茨矩阵，并建立了将其与对应块汉克尔矩阵关联的行列式恒等式；最后解决了关于赫维茨型矩阵多项式系数矩阵行列式正定性的猜想，证明该性质在次数至多为3时成立，但通过显式反例表明在次数为4时不成立。
英文摘要
Consider the matrix polynomial $\mathbf f_n(z)=I_qz^n+A_1z^{\,n-1}+\cdots+A_n,$ where $A_j\in\mathbb C^{q\times q}$. Write $\mathbf f_n(z)=\mathbf h_n(z^2)+z\,\mathbf g_n(z^2).$ A matrix polynomial $\mathbf f_n$ is called a Hurwitz-type matrix polynomial if, for $n=2m$, $\mathbf g_n(z)\mathbf h_n(z)^{-1}$, and for $n=2m+1$, $\mathbf h_n(z)\bigl(z\mathbf g_n(z)\bigr)^{-1}$, admit finite continued fraction expansions with positive definite coefficients. We derive explicit formulas for the coefficients of Hurwitz-type matrix polynomials in terms of orthogonal matrix polynomials, Markov parameters, and Schur complements. We introduce the associated block Hurwitz matrix and establish determinant identities relating it to the corresponding block Hankel matrices. Finally, we settle a conjecture concerning the positivity of the determinants of the coefficient matrices of Hurwitz-type matrix polynomials. We prove that this property holds for degrees at most three, but fails in degree four by means of an explicit counterexample.
