On the maximal numerical range of some matrices

Linear & Multilinear Algebra, 2020

Several new verifiable conditions are established for matrices of the form αI n−k C D βI k to have the numerical range equal the convex hull of at most k ellipses. For k = 2, these conditions are also necessary, provided that the ellipses are co-centered.

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, 2006

The following question is addressed: To what extent the n-tuple of m × m Hermitian matrices is determined by its joint numerical range? The cases m = 2, n arbitrary and m = n = 3 are considered in detail. (I.M. Spitkovsky).

Linear and Multilinear Algebra, 2012

Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3 × 3 and 4 × 4 matrices. In the latter case, it is proved that a 4 × 4 unitarily irreducible companion matrix cannot have 3 flat portions on the boundary of its numerical range. Numerical examples are given to illustrate the main results.

Linear Algebra and Its Applications, 1997

Let A be an n X n complex matrix. Then the numerical range of A, W(A), is defined to be {r*Ax : x E C", x*x = 1). In this article a series of tests is given, allowing one to determine the shape of W(A) for 3 X 3 matrices. Reconstruction of A, up to unitary similarity, from W(A) is also examined. 0 Elsevier Science Inc., 1997

Linear and Multilinear Algebra, 2019

We completely characterize the higher rank numerical range of the matrices of the form J n (α) ⊕ βI m , where J n (α) is the n × n Jordan block with eigenvalue α. Our characterization allows us to obtain concrete examples of several extreme properties of higher rank numerical ranges.

Linear and Multilinear Algebra, 1979

Let 1 ≤ m ≤ n, and let χ : H → C be a degree 1 character on a subgroup H of the symmetric group of degree m. The generalized matrix function on an m × σ(j) , and the decomposable numerical radius of an n × n matrix A on orthonormal tensors associated with χ is defined by r ⊥ χ (A) = max{|d χ (X * AX)| : X is an n × m matrix such that X * X = I m }.

Operators and Matrices, 2019

We show that the maximal numerical range of an operator has a non-empty intersection with the boundary of its numerical range if and only if the operator is normaloid. A description of this intersection is also given. First, let us set some notation and terminology. For a subset X of the complex plane C, by cl X, ∂X, and conv X we will denote the closure, boundary, and the convex hull of X, respectively. By an "operator" we throughout the paper understand a bounded linear operator acting on a Hilbert space H. The numerical range of such an operator A is defined by the formula W (A) = { Ax, x : x ∈ H, x = 1}, where .,. and. stand, respectively, for the scalar product on H and the norm associated with it. Introduced a century ago in the works by Toeplitz [8] and Hausdorrf [6] (and thus also known as the Toeplitz-Hausdorff set), it since has been a subject of intensive research. We mention here only [4] as a standard source of references, and note the following basic properties: Due to the Cauchy-Schwarz inequality, the set W (A) is bounded. Namely, w(A) := sup{|z| : z ∈ W (A)} ≤ A ; (1) w(A) is called the numerical radius of A.

Banach Journal of Mathematical Analysis, 2020

Let $$M_{n}({\mathbb {R}}_{+})$$ M n ( R + ) be the set of all $$n \times n$$ n × n nonnegative matrices. Recently, in Tavakolipour and Shakeri (Linear Multilinear Algebra 67, 2019, https://doi.org/10.1080/03081087.2018.1478946), the concept of the numerical range in tropical algebra was introduced and an explicit formula describing it was obtained. We study the isomorphic notion of the numerical range of nonnegative matrices in max algebra and give a short proof of the known formula. Moreover, we study several generalizations of the numerical range in max algebra. Let $$1 \le k \le n$$ 1 ≤ k ≤ n be a positive integer and $$C \in M_{ n}({\mathbb {R}}_{+}).$$ C ∈ M n ( R + ) . We introduce the notions of max $$k-$$ k - numerical range and max $$C-$$ C - numerical range. Some algebraic and geometric properties of them are investigated. Also, max numerical range $$W_\text {max}(\varSigma )$$ W max ( Σ ) of a bounded set $$\varSigma$$ Σ of $$n \times n$$ n × n nonnegative matrices is in...

2017

The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt&#39;s lemma and the Ky Fan&#39;s theorem on the block numerical range are extended.