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Solving Matrix Polynomial Equations
Cybernetics and Systems AnalysisMatrix equations and systems of matrix equations are widely used in problems of optimization of control systems, in mathematical economics. However, methods for solving them are developed only for the most popular matrix equations – the Riccati and Lyapunov equations, and there is no universal approach to solving problems of this class.
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2019
Given a square matrix \(M = (u_{ij})_{n \times n}\) and an m-order matrix polynomial \(f_m(M) = \sum _{k=0}^{m} a_k M^k = a_0 I + a_1M + a_2 M^2 + \cdots + a_m M^m\), if M is a dense matrix and is perturbed to become \(M'\) at a single entry, say \(u_{pq}\), a straightforward re-calculation of \(f_{m}(M')\) would require \(O(n^{\omega } \cdot \alpha (m)
Wei Ding, Ke Qiu
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Given a square matrix \(M = (u_{ij})_{n \times n}\) and an m-order matrix polynomial \(f_m(M) = \sum _{k=0}^{m} a_k M^k = a_0 I + a_1M + a_2 M^2 + \cdots + a_m M^m\), if M is a dense matrix and is perturbed to become \(M'\) at a single entry, say \(u_{pq}\), a straightforward re-calculation of \(f_{m}(M')\) would require \(O(n^{\omega } \cdot \alpha (m)
Wei Ding, Ke Qiu
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Numerical Range of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications, 1994Let \(A_ i\), \(i = 1, \dots, m\) be \(n \times n\) matrices with complex coefficients and consider the matrix polynomial \(P(\lambda) = \sum_{i=0}^ m A_ i \lambda^ i\). The numerical range of \(P(\lambda)\) is defined through \[ W \bigl( P(\lambda) \bigr) : = \{\mu \in \mathbb{C} \mid x^* P(\mu)x = 0 \quad \text{for some nonzero} \quad x \in \mathbb{C}
Li, Chi-Kwong, Rodman, Leiba
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1990
An overview is given of some classical and recent results concerning zeros of orthogonal matrix polynomials on the unit circle. The basic questions are: How these zeros are located in the complex plane? Conversely, what conditions on the location of the zeros of a given matrix polynomial ensure that the polynomial is orthogonal?
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An overview is given of some classical and recent results concerning zeros of orthogonal matrix polynomials on the unit circle. The basic questions are: How these zeros are located in the complex plane? Conversely, what conditions on the location of the zeros of a given matrix polynomial ensure that the polynomial is orthogonal?
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On solvents of matrix polynomials
Applied Numerical Mathematics, 2003The author considers the matrix equation \(X^m+A_1X^{m-1}+\ldots +A_m=0\) where the matrices \(A_i\) and \(X\) are complex and \(n\times n\). The author reviews the basic theory of matrix polynomials and the construction of solvents of such equations in terms of the concept of eigenpair.
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Polynomial and Matrix Near-Rings
Arabian Journal for Science and Engineering, 2011zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Computing the Invariant Polynomials of a Polynomial Matrix. II
Journal of Mathematical Sciences, 2004zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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