TY - JOUR
T1 - The Perturbation Analysis of the Product of Singular Vector Matrices $UV^T$
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 - <p style="text-align: justify;">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. <br/>This inequality is applicable to the computational error estimation of orthogonalization of a matrix, especially in the strapdown inertial navigation system.</p>