Pattern matrices are a special construction of matrices from a given function φ with the property that the singular values of the matrices correspond nicely to the Fourier coefficients of φ. Also, the eigenvalues “corresponding” to φ̂(S) have an inverse exponential dependence on the size of S. Hence, if φ has mass only on the high-degree Fourier coefficients, then the corresponding pattern matri...

Multiplicative backward stability results are presented for two algorithms which compute the singular value decomposition of dense matrices. These algorithms are the classical onesided Jacobi algorithm, with a stringent stopping criterion, and an algorithm which uses one-sided Jacobi to compute high accurate singular value decompositions of matrices given as rank-revealing factorizations. When ...

Notions of externally and internally positive singular discrete-time linear systems are introduced. It is shown that a singular discrete-time linear system is externally positive if and only if its impulse response matrix is non-negative. Sufficient conditions are established under which a single-output singular discrete-time system with matrices in canonical forms is internally positive. It is...

This report presents probability inequalities for sums of adapted sequences of random, self-adjoint matrices. The results frame simple, easily verifiable hypotheses on the summands, and they yield strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. The methods also specialize to sums of independent random matrices. 1. Main Results This technical report is...

two symmetric matrices, where one matrix has a restricted rank. A comparison of the two cases reveals a remarkable similarity between Eckhart-Young theorem and Ky Fan’s maximum principle. The extremum principles of Ky Fan consider sums and products of eigenvalues of Hermitian matrices. The third part of the talk derives extended rules for rectangular matrices. In this case singular values take ...