J. Comp. Math., 4 (1986), pp. 245-248.

The Perturbation Analysis Of The Product Of Singular Vector Matrices UV^T

Jian-Qin Mao 1

1 Beijing Insititute of Aeronautics and Astroautics, Beijing, China

Received 1985-4-22 Revised Online 2006-11-20


Let A be an $n\times n$ nonsingular real matrix,which has singular value decomposition $A=U\sum V^T$.Assume A is perturbed to $\tilde{A}$ and $\tilde{A}$ has singular value decomposition $\tilde{A}=\tilde{U}\tilde{\sum}\tilde{V}^T$.It is proved that $\|\tilde{U}\tilde{V}^T-UV^T\|_F\leq \frac{2}{\sigma_n}\|\tilde{A}-A\|_F$,where $\sigma_n$ is the minimum singular value of A;$\|\dot\|_F$ denotes the Frobenius norm and n is the dimension of A. This inequality is applicable to the computational error estimation of orthogonalization of a matrix,especially in the strapdown inertial navigation system.

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