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Volume 7, Issue 4
Perturbation Bounds for the Polar Factors

Yuan Chen & Ji-Guang Sun

J. Comp. Math., 7 (1989), pp. 397-401.

Published online: 1989-07

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  • Abstract

Let $A$, $\tilde{A}\in C^{m\times n}$, rank (A)=rank ($\tilde{A}$)=$n$. Suppose that $A=QH$ and $\tilde{A}=\tilde{Q}\tilde{H}$ are the polar decompositions of $A$ and $\tilde{A}$, respectively. It is proved that $$\|\tilde{Q}-Q\|_F\leq 2\|A^+\|_2\|\tilde{A}-A\|_F$$ and $$\|\tilde{H}-H\|_F\leq \sqrt{2}\|\tilde{A}-A\|_F$$ hold, where $A^+$ is the Moore-Penrose inverse of $A$, and $\| \|_2$ and $\| \|_F$ denote the spectral norm and the Frobenius norm, respectively.

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@Article{JCM-7-397, author = {Chen , Yuan and Sun , Ji-Guang}, title = {Perturbation Bounds for the Polar Factors}, journal = {Journal of Computational Mathematics}, year = {1989}, volume = {7}, number = {4}, pages = {397--401}, abstract = {

Let $A$, $\tilde{A}\in C^{m\times n}$, rank (A)=rank ($\tilde{A}$)=$n$. Suppose that $A=QH$ and $\tilde{A}=\tilde{Q}\tilde{H}$ are the polar decompositions of $A$ and $\tilde{A}$, respectively. It is proved that $$\|\tilde{Q}-Q\|_F\leq 2\|A^+\|_2\|\tilde{A}-A\|_F$$ and $$\|\tilde{H}-H\|_F\leq \sqrt{2}\|\tilde{A}-A\|_F$$ hold, where $A^+$ is the Moore-Penrose inverse of $A$, and $\| \|_2$ and $\| \|_F$ denote the spectral norm and the Frobenius norm, respectively.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9489.html} }
TY - JOUR T1 - Perturbation Bounds for the Polar Factors AU - Chen , Yuan AU - Sun , Ji-Guang JO - Journal of Computational Mathematics VL - 4 SP - 397 EP - 401 PY - 1989 DA - 1989/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9489.html KW - AB -

Let $A$, $\tilde{A}\in C^{m\times n}$, rank (A)=rank ($\tilde{A}$)=$n$. Suppose that $A=QH$ and $\tilde{A}=\tilde{Q}\tilde{H}$ are the polar decompositions of $A$ and $\tilde{A}$, respectively. It is proved that $$\|\tilde{Q}-Q\|_F\leq 2\|A^+\|_2\|\tilde{A}-A\|_F$$ and $$\|\tilde{H}-H\|_F\leq \sqrt{2}\|\tilde{A}-A\|_F$$ hold, where $A^+$ is the Moore-Penrose inverse of $A$, and $\| \|_2$ and $\| \|_F$ denote the spectral norm and the Frobenius norm, respectively.

Chen , Yuan and Sun , Ji-Guang. (1989). Perturbation Bounds for the Polar Factors. Journal of Computational Mathematics. 7 (4). 397-401. doi:
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