The numerical radius and specttural matrices

In this paper we investigate spectral matrices, i.e., matrices with equal spectral and numerical radii. Various characterizations and properties of these matrices are given.     

Quaternionic Numerical Range and Real Subspaces

Let W(A) be the numerical range of an n × n quaternionic matrix A and V a real subspace of the skew field of real quaternions. In this note the authors consider the relation among the shape of W(A), the convexity of V∩W(A): and the validity of the equality V∩W(A) = W_v(A), where W_v (A) is the orthogonal projection of W(A) into V.     

The numerical radius and bounds for zeros of a polynomial

Let be a monic polynomial. We obtain two bounds for zeros of via the Perron root and the numerical radius of the companion matrix of the polynomial.

     

Generalized numerical ranges, joint positive definiteness and multiple eigenvalues

We prove a convexity theorem on a generalized numerical range that combines and generalizes the following results: 1) Friedland and Loewy's result on the existence of a nonzero matrix with multiple first eigenvalue in subspaces of hermitian matrices, 2) Bohnenblust's result on joint positive definiteness of hermitian matrices, 3) the Toeplitz-Hausdorff Theorem on the convexity of the classical numerical range and its various generalizations by Au-Yeung, Berger, Brickman, Halmos, Poon, Tsing and Westwick.

     

Linear operators preserving the numerical range (radius) on triangular matrices

We characterize those linear operators on triangular or diagonal matrices preserving the numerical range or radius.     

The block numerical range of matrix polynomials

Block numerical ranges of matrix polynomials, especially the quadratic numerical range, are considered. The main results concern spectral inclusion, boundedness of the block numerical range, an estimate of the resolvent in terms of the quadratic numerical range, geometrical properties of the quadratic numerical range, and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions. As an application, we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials.     

The polynomial numerical hull of a matrix and algorithms for computing the numerical range

Let A be an n × n matrix. In this paper we discuss theoretical properties of the polynomial numerical hull of A of degree one and assemble them into three algorithms to computing the numerical range of A.     

Linear operators preserving the higher numerical radius of matrices

A linear operator T on a matrix space is said to be unital if T(I) = I. In this note we characterize the unital lineart operators on matrix spaces that preserve the k-numerical radius. Using the results obtained we easily determine the structure of all linear operators on the space of n × n complex matrices that preserve the k-numerical range. This completes the work of Pierce and Watking, who obtained the results for the cases when n ≠ n2k.     

Numerical radius inequalities for operator matrices

We prove a numerical radius inequality for operator matrices, which improves an earlier inequality due to Hou and Du. As an application of this numerical radius inequality, we derive a new bound for the zeros of polynomials.     

A new concept for block operator matrices:the quadratic numerical range

In this paper a new concept for 2×2-block operator matrices – the quadratic numerical range – is studied. The main results are a spectral inclusion theorem, an estimate of the resolvent in terms of the quadratic numerical range, factorization theorems for the Schur complements, and a theorem about angular operator representations of spectral invariant subspaces which implies e.g. the existence of solutions of the corresponding Riccati equations and a block diagonalization. All results are new in the operator as well as in the matrix case.     

On numerical ranges of the compressions of normal matrices

For an n × n normal matrix A, whose numerical range NR[A] is a k-polygon (k ? n), an n × (k − 1) isometry matrix P is constructed by a unit vector υ∈Cⁿ, and NR[P∗AP] is inscribed to NR[A]. In this paper, using the notations of NR[P∗AP] and some properties from projective geometry, an n × n diagonal matrix B and an n × (k − 2) isometry matrix Q are proposed such that NR[P∗AP] and NR[Q∗BQ] have as common support lines the edges of the k-polygon and share the same boundary points with the polygon. It is proved that the boundary of NR[P∗AP] is a differentiable curve and the boundary of the numerical range of a 3 × 3 matrix P∗AP is an ellipse, when the polygon is a quadrilateral.     

Matrix polynomials with spectral radius equal to the numerical radius

In this paper we study a class of matrix polynomials with the property that spectral radius and numerical radius coincide. Special attention is paid to the spectrum on the boundary of the numerical range.