NUMERICAL RANGE AND POSITIVE BLOCK MATRICES

Some results on the block numerical range

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's lemma and the Ky Fan's theorem on the block numerical range are extended.

Positive definiteness of tridiagonal matrices via the numerical range

Numerical Range for Random Matrices

We analyze the numerical range of high-dimensional random matrices, obtaining limit results and corresponding quantitative estimates in the non-limit case. For a large class of random matrices their numerical range is shown to converge to a disc. In particular, numerical range of complex Ginibre matrix almost surely converges to the disk of radius √ 2. Since the spectrum of non-hermitian random...

A Symmetric Numerical Range for Matrices

For each norm v on <en, we define a numerical range Z., which is symmetric in the sense that Z. =Z"D, where vD is the dual norm. We prove that, for aE <enn, 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...

Schur product of matrices and numerical radius (range) preserving maps

Let F (A) be the numerical range or the numerical radius of a square matrix A. Denote by A◦B the Schur product of two matrices A and B. Characterizations are given for mappings on square matrices satisfying F (A ◦ B) = F (φ(A) ◦ φ(B)) for all matrices A and B. Analogous results are obtained for mappings on Hermitian matrices. 2000 Mathematics Subject Classification. 15A04, 15A18, 15A60