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Numerical range of matrix polynomials. (English) Zbl 0814.15023

Let \(A_ i\), \(i = 1, \dots, m\) 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 \[ W \bigl( P(\lambda) \bigr) : = \{\mu \in \mathbb{C} \mid x^* P(\mu)x = 0 \quad \text{for some nonzero} \quad x \in \mathbb{C}^ n\}. \] Motivated by applications of overdamped vibration systems the authors investigate the relationship between the geometrical properties of \(W(P(\lambda))\) and the algebraic and analytic properties of \(P(\lambda)\).
After stating several basic properties the main results are given in Section 3. In this section new conditions are derived which guarantee the factorization of the matrix polynomial \(P(\lambda)\) into linear factors. The results indicate that the possible factorization of \(P(\lambda)\) into linear factors is closely related to topological properties of the numerical range \(W(P(\lambda))\).
In section 4 the paper concentrates on linear polynomial matrices with Hermitian coefficients. The paper concludes with a study of matrix polynomials having a “degenerate numerical range”.

MSC:

15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
15A22 Matrix pencils
15A23 Factorization of matrices
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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