On best approximations of polynomials in matrices in the matrix 2-norm

We show that certain matrix approximation problems in the matrix 2-norm have uniquely defined solutions, despite the lack of strict convexity of the matrix 2-norm. The problems we consider are generalizations of the ideal Arnoldi and ideal GMRES approximation problems introduced by Greenbaum and Trefethen [SIAM J. Sci. Comput., 15 (1994), pp. 359–368]. We also discuss general characterizations of best approximation in the matrix 2-norm and provide an example showing that a known sufficient condition for uniqueness in these characterizations is not necessary

On Chebyshev polynomials of matrices

The mth Chebyshev polynomial of a square matrix A is the monic polynomial that minimizes the matrix 2-norm of $p(A)$ over all monic polynomials $p(z)$ of degree m. This polynomial is uniquely defined if m is less than the degree of the minimal polynomial of A. We study general properties of Chebyshev polynomials of matrices, which in some cases turn out to be generalizations of well-known properties of Chebyshev polynomials of compact sets in the complex plane. We also derive explicit formulas of the Chebyshev polynomials of certain classes of matrices, and explore the relation between Chebyshev polynomials of one of these matrix classes and Chebyshev polynomials of lemniscatic regions in the complex plane

On Approximation Problems With Zero-Trace Matrices

In this paper we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms. We deal with special cases of approximation of a matrix by zero-trace matrices. Moreover, some characterizations of zero-trace matrices are given by means of matrix approximation problems. ------------------------------------------------------------------------------------------------------------------ 1. INTRODUCTION Let A = [a ij ] 2 C n\Thetan be a complex matrix. The trace of A is equal to tr(A) = X j a jj : It is well-known that tr(A) = 0 if and only if A is a commutator, that is, A = XY \Gamma Y X for some matrices X and Y . In this paper we consider some approximation problems, involving zero-trace matrices, with respect to an arbitrary unitarily invariant norm jj \Delta jj. A norm jj \Delta jj is unitarily invariant if jjUAjj = jjAV jj = jjAjj for all unitary matrices U and V . The most popular unitarily invariant norms are the c p -..