Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification
Hermitian matrix polynomials with real eigenvalues of definite type. Part I: Classification
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具有确定类型的实特征值的埃尔米特矩阵多项式。
DOI：
10.1016/j.laa.2010.08.035
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发表时间：
2012
影响因子：
1.1
通讯作者：
Al-Ammari M
中科院分区：
文献类型：
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作者：
Al-Ammari M
具有真实的特征值的厄米特矩阵多项式的谱性质已经被广泛研究，通过类，如确定的或可确定的铅笔，确定的，双曲的，或拟双曲的矩阵多项式，和过阻尼或陀螺稳定的二次型。我们给出了一个统一的治疗这些和相关的类，使用的特征值类型（或符号特征）作为一个共同的线程。每个类的等价条件以一致的格式给出。我们表明，这些类形成一个层次，所有这些都包含在新的类拟定矩阵多项式。除了收集和统一现有的结果，我们还做出了一些新的贡献。我们提出了一个新的双曲性的特征值类型的分布方面的真实的线。通过分析它们对特征值类型的影响，我们表明，齐次旋转允许结果的矩阵多项式与非奇异或明确的领导系数被转化为结果没有这样的要求的领导系数，这是重要的治疗明确和拟明确的多项式。我们还给出了拟双曲矩阵多项式与同阶的真实的对角拟双曲矩阵多项式严格等谱的一个充要条件，并证明了在二次情形下和对任何双曲矩阵多项式，这个条件总是满足的，从而确定了一类重要的新的可对角化矩阵多项式.
The spectral properties of Hermitian matrix polynomials with real eigenvalues have been extensively studied, through classes such as the definite or definitizable pencils, definite, hyperbolic, or quasihyperbolic matrix polynomials, and overdamped or gyroscopically stabilized quadratics. We give a unified treatment of these and related classes that uses the eigenvalue type (or sign characteristic) as a common thread. Equivalent conditions are given for each class in a consistent format. We show that these classes form a hierarchy, all of which are contained in the new class of quasidefinite matrix polynomials. As well as collecting and unifying existing results, we make several new contributions. We propose a new characterization of hyperbolicity in terms of the distribution of the eigenvalue types on the real line. By analyzing their effect on eigenvalue type, we show that homogeneous rotations allow results for matrix polynomials with nonsingular or definite leading coefficient to be translated into results with no such requirement on the leading coefficient, which is important for treating definite and quasidefinite polynomials. We also give a sufficient and necessary condition for a quasihyperbolic matrix polynomial to be strictly isospectral to a real diagonal quasihyperbolic matrix polynomial of the same degree, and show that this condition is always satisfied in the quadratic case and for any hyperbolic matrix polynomial, thereby identifying an important new class of diagonalizable matrix polynomials.