Matrices with Positive Definite Hermitian Part : Inequalities and Linear

The Hermitian and skew-Hermitian parts of a square matrix A are deened by H(A) (A + A)=2 and S(A) (A ? A)=2: We show that the function f(A) = (H(A ?1)) ?1 is convex with respect to the Loewner partial order on the cone of matrices with positive deenite Hermitian part. That is, for any matrices A and B with positive deenite Hermitian part ff(A) + f(B)g=2 ? f(fA + Bg=2) is positive semideenite: Using this basic fact we prove a variety of inequalities involving norms, Hadamard products and submatrices and a perturbation result for the function f. These results are generalizations of results for positive deenite matrices. Often the quantity H (A) kH(A ?1) ?1 k 2 kH(A) ?1 k 2 plays the role that 2 (A) kAk 2 kA ?1 k 2 plays in inequalities involving positive deenite matrices. (kk 2 denotes the spectral norm.) Finally we derive a bound on the backward and forward error in ^ x the solution to Ax = b with H(A) positive deenite (1) computed by Gaussian elimination without pivoting in nite precision. This result is analogous to Wilkinson's result for positive deenite matrices and gives a rigorous criterion for deciding when it is numerically safe not to pivot when solving (1). (R)) denote the space of n n complex (respectively, real) matrices. We call A 2 M n (C) positive deenite (respectively, positive semideenite) if A is Hermitian and x Ax > 0 (respectively, x Ax 0) for all nonzero x 2 C n. The Hermitian part of A is H(A) (A + A)=2 and the skew-Hermitian part of A is