Handbook of Linear Algebra, 2nd Edition

Chapter 25

Numerical Range

Chi-Kwong Li

College of William and Mary

The numerical range W(A) of an n × n complex matrix A is the collection of complex numbers of the form x*Ax, where x ∈ ℂn is a unit vector. It can be viewed as a “picture” of A containing useful information of A. Even if the matrix A is not known explicitly, the “picture” W(A) would allow one to “see” many properties of the matrix. For example, the numerical range can be used to locate eigenvalues, deduce algebraic and analytic properties, obtain norm bounds, help find dilations with simple structure, etc. Related to the numerical range are the numerical radius of A defined by w(A) = maxμ∈W(A) |μ| and the distance of W(A) to the origin denoted by $\tilde{w}\left(A\right)={\min }_{\mu \in W\left(A\right)}\left|\mu \right|$. If 0 ∉