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Volume 4, Issue 3
The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$

Jian-Qin Mao

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

Published online: 1986-04

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  • 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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@Article{JCM-4-245, author = {Jian-Qin Mao}, title = {The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$}, journal = {Journal of Computational Mathematics}, year = {1986}, volume = {4}, number = {3}, pages = {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.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9585.html} }
TY - JOUR T1 - The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$ AU - Jian-Qin Mao JO - Journal of Computational Mathematics VL - 3 SP - 245 EP - 248 PY - 1986 DA - 1986/04 SN - 4 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9585.html KW - AB -

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.

Jian-Qin Mao. (1986). The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$. Journal of Computational Mathematics. 4 (3). 245-248. doi:
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