Re: [R] SVD of a variance matrix

From: Ravi Varadhan <rvaradhan_at_jhmi.edu>
Date: Wed, 16 Apr 2008 13:37:29 -0400

No. The relationship U = V doesn't have to hold for positive-semidefinite matrices, just as it doesn't for an indefinite matrix (i.e. a matrix with both positive and negative eigenvalues), since you can have u_i = +/- (v_i )^T corresponding to the zero eigenvalue.

You may refer to, for example, GW Stewart's Matrix Algorithms (vol 1. Basic decompositions, SIAM 1998), page 70.

Ravi.



Ravi Varadhan, Ph.D.

Assistant Professor, The Center on Aging and Health

Division of Geriatric Medicine and Gerontology

Johns Hopkins University

Ph: (410) 502-2619

Fax: (410) 614-9625

Email: rvaradhan_at_jhmi.edu

Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html  



-----Original Message-----
From: Giovanni Petris [mailto:GPetris_at_uark.edu] Sent: Tuesday, April 15, 2008 8:07 PM
To: rvaradhan_at_jhmi.edu
Cc: rvaradhan_at_jhmi.edu; r-help_at_r-project.org Subject: Re: [R] SVD of a variance matrix

Hi Ravi,

Thank you for your useful reply. Does the result also hold for variance-covariance matrices that have one or more zero eigenvalues? Do you have a reference to suggest?

Thank you again!

Giovanni

> Date: Tue, 15 Apr 2008 18:14:11 -0400
> From: Ravi Varadhan <rvaradhan_at_jhmi.edu>
> Thread-index: AcifQeEz9B1geo3TQyesYlQGMCSuNgAAWF1QAACa9sA=
>
> Let me correct my reply a bit.
>
> U and V will differ by a factor of (-1) corresponding to negative
> eigenvalues (if any) of a general symmetric A. However, for symmetric
> positive-definite matrices (e.g. variance-covariance matrix), they will be
> identical.
>
> Ravi.
>
>



> -------
>
> Ravi Varadhan, Ph.D.
>
> Assistant Professor, The Center on Aging and Health
>
> Division of Geriatric Medicine and Gerontology
>
> Johns Hopkins University
>
> Ph: (410) 502-2619
>
> Fax: (410) 614-9625
>
> Email: rvaradhan_at_jhmi.edu
>
> Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html
>
>
>
>


> --------
>
>
> -----Original Message-----
> From: r-help-bounces_at_r-project.org [mailto:r-help-bounces_at_r-project.org]
On
> Behalf Of Ravi Varadhan
> Sent: Tuesday, April 15, 2008 6:03 PM
> To: 'Giovanni Petris'; r-help_at_r-project.org
> Subject: Re: [R] SVD of a variance matrix
>
> Yes. SVD of any symmetric (which is, of course, also square) matrix will
> always have U = V. Also, SVD is the same as spectral decomposition, and
the
> columns of U and V are the eigenvectors, but the singular values will be
the
> absolute value of eigenvalues.
>
> Ravi.
>
>


> -------
>
> Ravi Varadhan, Ph.D.
>
> Assistant Professor, The Center on Aging and Health
>
> Division of Geriatric Medicine and Gerontology
>
> Johns Hopkins University
>
> Ph: (410) 502-2619
>
> Fax: (410) 614-9625
>
> Email: rvaradhan_at_jhmi.edu
>
> Webpage: http://www.jhsph.edu/agingandhealth/People/Faculty/Varadhan.html
>
>
>
>


> --------
>
> -----Original Message-----
> From: r-help-bounces_at_r-project.org [mailto:r-help-bounces_at_r-project.org]
On
> Behalf Of Giovanni Petris
> Sent: Tuesday, April 15, 2008 5:43 PM
> To: r-help_at_r-project.org
> Subject: [R] SVD of a variance matrix
>
>
> Hello!
>
> I suppose this is more a matrix theory question than a question on R,
> but I will give it a try...
>
> I am using La.svd to compute the singular value decomposition (SVD) of
> a variance matrix, i.e., a symmetric nonnegative definite square
> matrix. Let S be my variance matrix, and S = U D V' be its SVD. In my
> numerical experiments I always got U = V. Is this necessarily the
> case? Or I might eventually run into a SVD which has U != V?
>
> Thank you in advance for your insights and pointers.
>
> Giovanni
>
> --
>
> Giovanni Petris <GPetris_at_uark.edu>
> Associate Professor
> Department of Mathematical Sciences
> University of Arkansas - Fayetteville, AR 72701
> Ph: (479) 575-6324, 575-8630 (fax)
> http://definetti.uark.edu/~gpetris/
>
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https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code. Received on Wed 16 Apr 2008 - 17:45:04 GMT

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