Numerical Range of Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
Differential calculus for some p-norms of the fundamental matrix with applications
Journal of Computational and Applied Mathematics
The Numerical Range of Self-Adjoint Quadratic Matrix Polynomials
SIAM Journal on Matrix Analysis and Applications
The Quadratic Eigenvalue Problem
SIAM Review
| Hi-index | 7.29 |
Let A, B and C be three n × n nonzero Hermitian matrices. The triple (A,B,C) is called defnite if the convex hull of the joint numerical range F(A,B,C)={(x*Ax, x*Bx, x*Cx) ∈ R3: x ∈ Cn,x*x=1} does not contain (0,0,0). If the triple (A,B,C) is nondefinite, then the numerical ranges of the matrix polynomials Q(λ)= Aλv3 + Bλv2 + Cλv1 (v3 v2 v1 ≥ 0) and L(λ) = Aλξ2 +(B + iC)λξ1 (ξ2 ξ1 ≥ 0) coincide with the whole complex plane, providing no information. As a consequence, it is of particular interest to characterize a definite triple (A,B,C) and find the distance between (0,0,0) and the boundary F(A,B,C). The distance between a nondefinite triple (A,B,C) and the "nearest" definite triples with specified properties is also investigated. Moreover, applications of definite triples on matrix polynomials of special interest are presented.