Recently, there has been a significant interest in the combinatorial properties the ring of n×n matrices. The aim of this note is to describe a short (may be the shortest possible) presentation of the matrix ring Matn(Z). This presentation is significantly shorter than the previously known ones, see [7]. Surprisingly, the number of relations in the presentation does not depend on the size of th...

The Schur method for computing a matrix square root reduces the matrix to the Schur triangular form and then computes a square root of the triangular matrix. We show that by using a recursive blocking technique the computation of the square root of the triangular matrix can be made rich in matrix multiplication. Numerical experiments making appropriate use of level 3 BLAS show significant speed...

Given a complex matrix H, we consider the decomposition H = QRP∗, where R is upper triangular and Q and P have orthonormal columns. Special instances of this decomposition include the singular value decomposition (SVD) and the Schur decomposition where R is an upper triangular matrix with the eigenvalues of H on the diagonal. We show that any diagonal for R can be achieved that satisfies Weyl’s...