David Wenzel’s research while affiliated with Chemnitz University of Technology and other places

The values of the smallest possible constant C in the inequality ∥XY -YX∥ = C∥X∥∥Y∥ on the space of real or complex n×n-matrices are investigated for different norms. Mathematics Subject Classification (2000)Primary–15A45–Secondary–15A69

We determine the sharpest constant $C_{p,q,r}$ such that for all complex matrices X and Y, and for Schatten p-, q- and r-norms the inequality $\|XY-YX\|_p\leq C_{p,q,r}\|X\|_q\|Y\|_r$ is valid. The main theoretical tool in our investigations is complex interpolation theory.

It is known that for any nonzero complex n×n matrices X and Y the quotient of Frobenius norms‖XY-YX‖F‖X‖F‖Y‖Fdoes not exceed 2. However, except for some special cases, only necessary conditions for attaining this bound have been found so far. We will completely characterize the pairs of matrices that satisfy equality with the quotient’s maximum.

In an earlier paper we conjectured an inequality for the Frobenius norm of the commutator of two matrices. This conjecture was recently proved by Seak-Weng Vong and Xiao-Qing Jin. We here give a completely different proof of this inequality, prove some related results, and embark on the corresponding question for unitarily invariant norms.

This paper is motivated by recent studies of Huang et al. on distributed PCA and network anomaly detection and contains a rigorous derivation of bounds for the expected value and the variance of the spectral norm of the error in large covariance matrices. This derivation is based on a deep result by Yin et al. (Probab. Theor. Relat. Fields 1988; 78:509–521), which gives the asymptotics of the maximal eigenvalue of a random matrix as the matrix dimension goes to infinity. Copyright © 2007 John Wiley & Sons, Ltd.

Testing whether a given matrix is a Toeplitz-plus-Hankel matrix amounts to the verification of a system of linear equations for the matrix entries. If the matrix dimension is large, we are forced to work with the computer and hence cannot check whether something is exactly zero. We provide bounds such that if a test quantity is smaller than the bound, then the system of linear equations may be accepted to be valid and the probability for erroneously accepting the validity of the system is smaller than a prescribed value.

If the matrix of a square linear system is nonsingular but has very small singular values, then tiny perturbations of the right-hand side may cause drastic changes in the solution. We show that the probability for this to happen is very close to zero if sufficiently many singular values of the matrix are bounded away from zero.

Numerical experiments show that the Frobenius norm of the commutator of two large matrices typically clusters sharply around a certain value, which, moreover, is much smaller than one would predict. The purpose of this paper is to give a rigorous foundation of this phenomenon. We also discuss the question how big the Frobenius norm of commutator of the two matrices can actually be.