Matrices with Positive Definite Hermitian Part: Inequalities and Linear Systems
Roy Mathias · SIAM Journal on Matrix Analysis and Applications · 1992
The Hermitian and skew-Hermitian parts of a square matrix A are defined by \[ H( A ) \equiv ( A + A^ * ) /2\qquad {\text{and}}\qquad S ( A ) \equiv ( A - A^ * )/2. \]It is shown 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 definite Hermitian part. That is, for any matrices A and B with positive definite Hermitian part \[ \{ f ( A ) + f ( B ) \}/2 - f ( \{ A + B \} /2 )\quad \text{is positive semidefinite}. \] Using this basic fact, this paper proves 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 definite matrices. Often the quantity \[ \kappa _H (A) \equiv \left\| H(A^{ - 1} )^{ - 1} \right\|_2 \left\| H(A)^{ - 1} \right\|_2 \] plays the role that $\kappa _2 (A) \equiv \| A \|_2 \| A^{ - 1} \|_2 $ plays in inequalities involving positive definite matrices. ($\| \cdot \|_2 $ denotes the spectral norm.) Finally a bound is derived on the backward and forward error in $\hat x$, the solution to \[ (1) \qquad Ax = b\quad {\text{ with }}\,H(A)\,{\text{positive definite}} \] computed by Gaussian elimination without pivoting in finite precision. This result is analogous to Wilkinson’s result for positive definite matrices and gives a rigorous criterion for deciding when it is numerically safe not to pivot when solving (1).