Lecture notes on Numerical Range Draft on
نویسنده
چکیده
The numerical range W (A) of an n×n matrix A is the collection of complex numbers of the form x∗Ax, where x ∈ C 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 w̃(A) = minμ∈W (A) |μ|. The quantities w(A) and w̃(A) are useful in studying perturbation, convergence, stability, approximation problems. Basic results in our discussion can be found in [23], [29, Chapter 22], [32, Chapter 1], and [34, Chapter 6]. In addition, we will mention some immediately related papers and books containing more related work and references that readers can further pursue.
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