New Perturbation Bounds for the Unitary Polar Factor

Ren-Cang Li

EECS Department, University of California, Berkeley

Technical Report No. UCB/CSD-94-852

, 1994

http://www2.eecs.berkeley.edu/Pubs/TechRpts/1994/CSD-94-852.pdf

Let A be an m x n (m >= n) complex matrix. It is known that there is a unique polar decomposition A = QH, where Q*Q = I, the n x n identity matrix, and H is positive definite, provided A has full column rank. This note addresses the following question: how much may Q change if A is perturbed? For the square case m = n our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's (SIAM J. Matrix Anal. Appl., 14 (1993), 588-597.). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.

BibTeX citation:

@techreport{Li:CSD-94-852,
    Author= {Li, Ren-Cang},
    Title= {New Perturbation Bounds for the Unitary Polar Factor},
    Year= {1994},
    Month= {Dec},
    Url= {http://www2.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html},
    Number= {UCB/CSD-94-852},
    Abstract= {Let <i>A</i> be an <i>m</i> x <i>n</i> (<i>m</i> >= <i>n</i>) complex matrix. It is known that there is a unique polar decomposition <i>A</i> = <i>QH</i>, where <i>Q</i>*<i>Q</i> = <i>I</i>, the <i>n</i> x <i>n</i> identity matrix, and <i>H</i> is positive definite, provided <i>A</i> has full column rank. This note addresses the following question: how much may <i>Q</i> change if <i>A</i> is perturbed? For the square case <i>m</i> = <i>n</i> our bound, which is valid for any unitarily invariant norm, is sharper and simpler than Mathias's (<i>SIAM J. Matrix Anal. Appl., <b>14</b> (1993), 588-597.</i>). For the non-square case, we also establish a bound for unitarily invariant norm, which has not been done in literature.},
}

EndNote citation:

%0 Report
%A Li, Ren-Cang 
%T New Perturbation Bounds for the Unitary Polar Factor
%I EECS Department, University of California, Berkeley
%D 1994
%@ UCB/CSD-94-852
%U http://www2.eecs.berkeley.edu/Pubs/TechRpts/1994/5883.html
%F Li:CSD-94-852