The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$

doi:1986-JCM-9585

The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$

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Year: 1986

Journal of Computational Mathematics, Vol. 4 (1986), Iss. 3 : pp. 245–248

Abstract

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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Journal Article Details

Publisher Name: Global Science Press

Language: English

DOI: https://doi.org/1986-JCM-9585

Journal of Computational Mathematics, Vol. 4 (1986), Iss. 3 : pp. 245–248

Published online: 1986-01

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Pages: 4

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