Bounds on condition number of a singular matrix and its applications

For each vector norm parallel toxparallel to, a matrix A has its operator norm parallel toAparallel to = max(parallel toxparallel tonot equal0) parallel toAxparallel to/parallel toxparallel to. If A is nonsingular, we can define the condition number of A, P(A) = parallel toAparallel to parallel toA(-1)parallel to. Let U be the set of the whole of norms defined on C-n. Huang [J. Computat. Math. 2 (1984) 356] shows that for a nonsingular matrix A is an element of C-nxn, there is no finite upper bound of P(A) while parallel to.parallel to varies on U if A not equal alphaI, and inf(parallel to.parallel tois an element ofU) parallel toAparallel to parallel toA(-1)parallel to = rho(A)rho(A(-1)). The first part of this paper will show that for a singular matrix A, we can have the same result on the sense of Drazin inverse of A. where rho(A) denotes the spectral radius of A. On the other hand, the SVD of a matrix can show the extension of the approach between the given matrix and a matrix whose rank is lower than its. In the second part, we can prove when a matrix is diagonalizable, and it's Jordan canonical form is A = PDiag(lambda(1), ..., lambda(n))P-1, then in the sense of P-norm, we can have a similar result. When Index(A) = 1, A(D) is the group inverse of A(g). In the third part we will prove a minimum property of group inverse. (C) 2004 Elsevier Inc. All rights reserved.