[R] SVD of a variance matrix

[Ravi Varadhan]

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/
> 
> ______________________________________________
> R-help at r-project.org mailing list
> 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.
> 
> ______________________________________________
> R-help at r-project.org mailing list
> 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.
>