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ALLSTAT  December 2007

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Subject:

Re: QUERY: relation between Eigenvalue Theory and Singularvalue Decomposition

From:

John Bibby <[log in to unmask]>

Reply-To:

John Bibby <[log in to unmask]>

Date:

Sat, 1 Dec 2007 13:21:12 -0000

Content-Type:

text/plain

Parts/Attachments:

Parts/Attachments

text/plain (107 lines)

Dear Stefano

But beware - your statements state EXISTENCE, not UNIQUENESS. (As you
indicate where you say "U and V may differ by signs of columns" - but this
is not the whole story).

Two references that you may find useful:

1. The Geometry of Generalized Inverses by William Kruskal
Journal of the Royal Statistical Society. Series B (Methodological), Vol.
37, No. 2 (1975), pp. 272-283
	
http://links.jstor.org/sici?sici=0035-9246(1975)37%3A2%3C272%3ATGOGI%3E2.0.C
O%3B2-C

2. We sweated over this for the Matrix Appendix of
Multivariate Analysis (Probability and Mathematical Statistics) (Probability
and Mathematical Statistics) by K. V. Mardia (Author), J. T. Kent (Author),
J. M. Bibby (Author) 
	
http://www.amazon.com/Multivariate-Analysis-Probability-Mathematical-Statist
ics/dp/0124712525

and I think that is fairly complete (but dated). A lot of it is in C.R.Rao's
"Linear Statistical Inference" but I think that has a few errors in it - the
German edition is corrected I believe, but that may not be very helpful.

In the 70's there where several books on generalized inverses which I
remember helped me turn the corner - see Kruskal for details.

PS: The Wikipedia article on generalizewd inverses is pretty weak - does
anyone have time to improve it?
http://en.wikipedia.org/wiki/Generalized_inverse


     JOHN BIBBY  aa42/MatheMagic
PS: Just been away. If any emails unanswered, please re-send.. 

All statements are on behalf of aa42.com Limited, a company wholly owned by
John Bibby and Shirley Bibby. See www.aa42.com/mathemagic and
www.mathemagic.org


-----Original Message-----
From: A UK-based worldwide e-mail broadcast system mailing list
[mailto:[log in to unmask]] On Behalf Of Stefano Sofia
Sent: 28 November 2007 16:52
To: [log in to unmask]
Subject: Re: QUERY: relation between Eigenvalue Theory and Singularvalue
Decomposition

Dear allstat users,
here there is a summary about the query reported at the bottom. 
I have to thank Prof. Alistair Watson for his help. 

Case of M real symmetric
Then there is an orthogonal matrix Q such that 

Q^T M Q = D,

where D is a diagonal matrix containing the eigenvalues.
For the SVD of M, there are orthogonal matrices U, V such that

M = U^T S V

where 
S contains the singular values (these are the moduli of the elements of D);
the matrices U, V are orthogonal (U and V may differ by signs of columns).

Case of M real but not necessarily symmetric
In this case the eigenvalues of M may not be real. There may not be an
orthogonal matrix of eigenvectors.

All one can say is that if the SVD is

M= U S V^T 

then

M^TM = V S^2 V^T

that is the eigenvalues of M^TM are the squares of the singular values of M,
and the matrix of eigenvectors of M^TM is V.


ORIGINAL QUESTION:
> Dear  allstat users,
> here there is a question about matrix algebra.
>
> Given a squared matrix M, according to eigenvalues theory (or eigenvalue
decomposition), I can find a matrix K such that J = KMK^{T} is diagonal,
with K^{T} being the transpose of K.
>
> Another important factorisation for M is called Singular Value
Decomposition (SVD). In this case M = USV^{*} where
> U is a unitary matrix, S is a matrix with nonnegative numbers on the
diagonal and zero off the diagonal and V^{*} denotes the conjugate transpose
of V, a unitary matrix.
>
> I think that in some cases, and in particular case of M being Hermitian,
these two factorisations are strictly related.
> Could anyone explain to me briefly which is the relation between these two
decompositions?
>
> thank you
> Stefano Sofia PhD

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