Bounds on condition number of singular matrix
Zhiping Xiong · Filomat · 2014
For each vector norm ||x||v, a matrix A ? Cmxn has its operator norm ||A||?v = maxx?O ||Ax||?/||x||v. If A is nonsingular, we can define the condition number of A ? Cnxn as P(A) = ||A||vv ||A-1||vv. If A is singular, the condition number of matrix A ? Cmxn may be defined as P+(A)=||A||?v ||A+||v?. Let U be the set of the whole self-dual norms. It is shown that for a singular matrix A ? Cmxn, there is no finite upper bound of P+(A), while ||.|| varies on U. On the other hand, it is shown that inf ||.||? U ||A||?v ||A+||v? = ?1(A)/?r(A), where ?1(A) and ?r(A) are the largest and smallest nonzero singular values of A, respectively.