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 normed linear spaces of matrices and show on an example that a known sufficient condition for uniqueness in these characterizations is not necessary.