Matrix Polynomials with Completely Prescribed Eigenstructure
Matrix Polynomials with Completely Prescribed Eigenstructure
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DOI：
10.1137/140964138
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发表时间：
2015-03
期刊：
影响因子：
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通讯作者：
F. Terán;F. Dopico;P. Dooren
中科院分区：
文献类型：
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作者：
F. Terán;F. Dopico;P. Dooren
我们提出了一个矩阵多项式存在的必要和充分条件时，它的程度，它的有限和无限的基本因子，其左，右最小指标规定。这些条件适用于任意无限域，主要由“指数和定理”确定，这是任何矩阵多项式的秩、次数、所有部分重数之和与所有最小指数之和之间的基本关系。这种多项式的存在性的证明是建设性的，因此，解决了一个非常一般的矩阵多项式与规定的完整的特征结构的逆问题。这个结果使我们能够解决一个给定的矩阵多项式的$\ell$-的存在性问题，以及确定其所有可能的大小和特征结构。
We present necessary and sufficient conditions for the existence of a matrix polynomial when its degree, its finite and infinite elementary divisors, and its left and right minimal indices are prescribed. These conditions hold for arbitrary infinite fields and are determined mainly by the “index sum theorem,” which is a fundamental relationship between the rank, the degree, the sum of all partial multiplicities, and the sum of all minimal indices of any matrix polynomial. The proof developed for the existence of such polynomial is constructive and, therefore, solves a very general inverse problem for matrix polynomials with prescribed complete eigenstructure. This result allows us to fix the problem of the existence of $\ell$-ifications of a given matrix polynomial, as well as to determine all their possible sizes and eigenstructures.