Numerical Range of Matrix Polynomials
Chi-Kwong Li, Leiba Rodman · SIAM Journal on Matrix Analysis and Applications · 1994
Let $M_n $ be the algebra of all $n \times n$ complex matrices. Suppose \[ P( \lambda ) = A_m \lambda ^m + A_{m - 1} \lambda ^{m - 1} + \cdots + A_0 \] is a matrix polynomial, where $A_i \in M_n $ and $\lambda $ is a complex variable. The numerical range of $P( \lambda )$ is defined as \[ W( P( \lambda ) ) = \{ \mu \in \mathbb{C}:x^ * P ( \lambda )x = 0\,{\text{ for some nonzero }}\,x \in \mathbb{C}^n \}. \] The numerical range of matrix polynomials has important applications to overdamped vibration systems with finite number of degrees of freedom and it is also related to stability theory. In this paper, the subject is studied systematically. The emphasis is on the relationship between the geometrical properties of $W( P( \lambda ) )$ and the algebraic and analytic properties of $P( \lambda )$. A factorization result, based on geometric properties of $W( P( \lambda ) )$ for certain classes of matrix polynomials with not necessarily hermitian coefficients is proved, and the set $W( P( \lambda ) )$ for a linear polynomial with hermitian matrices as coefficients is studied in detail. The results indicate that the information on $W( P( \lambda ) )$ is very useful in understanding matrix polynomials and also reflects the fact that it is highly nontrivial to give a complete description of the set $W( P( \lambda ) )$.