Bounds on condition number of a singular matrix and its applications

摘要

For each vector norm ∥x∥, a matrix A has its operator norm ∥A∥=max∥x∥≠0∥Ax∥∥x∥. If A is nonsingular, we can define the condition number of A, P(A)=∥A∥∥A−1∥. Let U be the set of the whole of norms defined on Cn. Huang [J. Computat. Math. 2 (1984) 356] shows that for a nonsingular matrix A∈Cn×n, there is no finite upper bound of P(A) while ∥·∥ varies on U if A≠αI, and inf∥·∥∈U∥A∥∥A−1∥=ρ(A)ρ(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 ρ(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(λ1,…,λn)P−1, then in the sense of P-norm, we can have a similar result.When Index(A)=1, AD is the group inverse of Ag. In the third part we will prove a minimum property of group inverse.