From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Orthogonal transformation
Date: 12 Dec 1999 17:29:48 -0500
Newsgroups: sci.math,sci.stat.math
Keywords: comparison of various matrix factorizations
In article <3852B948.2A6653D1@ucla.edu>,
Prabha Siddarth wrote:
:Is there an orthogonal transformation that transforms a symmetric
:positive definite matrix to an identity matrix?
:
:Thanks for any help,
:
You probably merged several true facts into one confusion (real mathices
seem to be the context):
(1) Every positive definite matrix A can be written as A = C * C'
(C' being the transpose of C), where C is invertible (can be
stipulated to be upper triangular)
This requires finding square roots, so some theories settle for
(2) Every positive definite matrix A can be written as A = L * D * L'
where D is diagonal with positive entries and L is invertible (can be
stipulated to be upper triangular with unit diagonal)
Here L and D can have entries from the same field where entries of A come
from (e.g. the rationals).
Facts (1) and (2) are related to "congruence of matrices": two matrices
A, B are called congruent if A = P * B * P' where P is invertible. The
statements say that every positive definite matrix is congruent to I, or
over the same field, at least to a diagonal p.d. matrix.
(It's a part of a more general statement about "inertia of symmetric
matrices".)
So far, orthogonality has not been mentioned.
With orthogonal matrices,
(3) every p.d. matrix A can be written as
A = U * D * U'
with U orthogonal and D diagonal with positive diagonal entries.
The field where the entries of D and U come from must contain roots of
characterictic polynomials of symmetric matrices.
(BTW, is it the same as the real part of the field of algebraic numbers,
if the entries of A are restricted to be rational?)
And you cannot do any better in general (forcing D to be I), as other
respondents wrote.
Cheers, ZVK(Slavek).