The numerical range of 3 × 3 matrices

Linear Algebra and its Applications, 2018

The joint numerical range W (F) of three hermitian 3-by-3 matrices F = (F 1 , F 2 , F 3) is a convex and compact subset in R 3. We show that W (F) is generically a three-dimensional oval. Assuming dim(W (F)) = 3, every one-or two-dimensional face of W (F) is a segment or a filled ellipse. We prove that only ten configurations of these segments and ellipses are possible. We identify a triple F for each class and illustrate W (F) using random matrices and dual varieties.

Linear and Multilinear Algebra, 1996

Let A, C be n × n complex matrices. We prove in the affirmative the conjecture that the C-numerical range of A, defined by W C (A) = {tr (CU * AU) : U is unitary} , is always star-shaped with respect to star-center (tr A)(tr C)/n. This result is equivalent to that the image of the unitary orbit {U * AU : U is unitary} of A under any complex linear functional is always star-shaped.

Electronic Journal of Linear Algebra, 2018

The maximal numerical range W0(A) of a matrix A is the (regular) numerical range W (B) of its compression B onto the eigenspace L of A*A corresponding to its maximal eigenvalue. So, always W0(A) ⊆ W (A). Conditions under which W0(A) has a non-empty intersection with the boundary of W (A) are established, in particular, when W0(A) = W(A). The set W0(A) is also described explicitly for matrices unitarily similar to direct sums of 2-by-2 blocks, and some insight into the behavior of W0(A) is provided when L has codimension one.

Opuscula Mathematica, 2011

A presentation of numerical ranges for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Further, we extend to the q-numerical range.

Numerische Mathematik, 1976

For each norm v on <en, we define a numerical range Z., which is symmetric in the sense that Z. =Z"D, where v D is the dual norm. We prove that, for aE <e nn , Z.(a) contains the classical field of values V(a). In the special case that v is the lcnorm, Z.(a) is contained in a set G(a) of Gershgorin type defined by C. R. Johnson. When a is in the complex linear span of both the Hennitians and the v-Hennitians, then Z.(a), V(a) and the convex hull of the usual v-numerical range V.(a) all coincide. We prove some results concerning points of V(a) which are extreme points of Z.(a).

Linear Algebra and its Applications, 1999

In [18], among other equivalent conditions, it is proved that a square complex matrix A is permutationally similar to a block-shift matrix if and only if for any complex matrix B with the same zero pattern as A, W (B), the numerical range of B, is a circular disk centered at the origin. In this paper, we add a long list of further new equivalent conditions. The corresponding result for the numerical range of a square complex matrix to be invariant under a rotation about the origin through an angle of 2π/m, where m 2 is a given positive integer, is also proved. Many interesting by-products are obtained. In particular, on the numerical range of a square nonnegative matrix A, the following unexpected results are established: (i) when the undirected graph of A is connected, if W (A) is a circular disk centered at the origin, then so is W (B), for any complex matrix B with the same zero pattern as A; (ii) when A is irreducible, if λ is an eigenvalue in the peripheral spectrum of A that lies on the boundary of W (A), then λ is a sharp point of W (A). We also obtain results on the numerical range of an irreducible square nonnegative matrix, which strengthen or clarify the work of Issos [9] and Nylen and Tam [14] on this topic. Open questions are posed at the end.

Computers & Mathematics with Applications, 1996

Through the linearization of a matrix polynomial P(A), the symmetry and the sharp points of the numerical range w(P(A)) are studied.

Linear & Multilinear Algebra, 2020

A complete description of 4-by-4 matrices αI C D βI , with scalar 2-by-2 diagonal blocks, for which the numerical range is the convex hull of two nonconcentric ellipses is given. This result is obtained by reduction to the leading special case in which C − D * also is a scalar multiple of the identity. In particular cases when in addition α−β is real or pure imaginary, the results take an especially simple form. An application to reciprocal matrices is provided.

Electronic Journal of Linear Algebra, 2013

A subset S of the complex plane has n-fold symmetry about the origin (n-sato) if z ∈ S implies e 2π n z ∈ S. The 3 × 3 matrices A for which the numerical range W (A) has 3-sato have been characterized in two ways. First, W (A) has 3-sato if and only if the spectrum of A has 3-sato while tr(A 2 A *) = 0. In addition, W (A) has 3-sato if and only if A is unitarily similar to an element of a certain family of generalized permutation matrices. Here it is shown that for an n × n matrix A, if a specific finite collection of traces of words in A and A * are all zero, then W (A) has n-sato. Further, this condition is shown to be necessary when n = 4. Meanwhile, an example is provided to show that the condition of being unitarily similar to a generalized permutation matrix does not extend in an obvious way.

1995

If we regard C as the matrix representation of a linear operator acting on an n-dimensional Hilbert space, then ~'(C) is the collection of all matrix representations of the same operator with respect to different orthonormal bases, and it is very useful in studying the operator. Goldberg and Straus [3] introduced the concept of C-numerical range of A •Jr', defined by Wc(A) = {tr(AX): X • ~(C)} = {tr(CX): X • ~'(A)}. Note that .*~ is equipped with the usual inner product (X, Y) = tr(XY*), and every linear functional f on .*¢" can be represented as f(X) = (X, F) for some suitable F •.*¢" depending on f. Thus the set Wc(A) can be regarded as the range of a linear functional on iX(C). The C-numerical range has been extensively studied, and many interesting results have been obtained, especially when C is hermitian (e.g., see [3], [8], [15], [19], [21], and their references).