From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik)
Subject: Re: Real Polar Decomposition
Date: 3 Feb 1999 16:08:27 -0500
Newsgroups: sci.math
Keywords: factoring real matrix as (symmetric pos. semidefinite)*(orthogonal)
In article <36B84C73.6001192A@e-technik.uni-ulm.de>,
Bernd Tibken wrote:
:Hello all,
:
:I would like to have a reference to the real polar decomposition which
:states that every real n*n matrix A has a representation A=H*U where H
:is symmetric and positive semidefinite and U is orthogonal. In the
:complex case with H hermitian and positive definite and U unitary I have
:a lot of references but unfortunately I need the result for the real
:case.
:
:Thank you for all your efforts to help me.
The procedure for complex matrices can be adjusted to end up with real
matrices if the input is real. Here is a sledgehammer approach (cracking a
walnut with a steamroller):
The singular value decomposition of a real matrix A is
A = U * S * V' (V' means the transpose of V)
where U, V are real orthogonal and S is diagonal, with entries
non-negative and sorted in descending order along the diagonal.
The matrices U and V will be real orthogonal because the columns of U are
the normalized, orthogonalized if necessary, eigenvectors of A*A', and
similarly for the columns of V related to A'*A.
Now comes the disappointingly simple result: Set
Q = U * V' and H = U * S * U'
and verify that Q is real orthogonal, and H is real non-negative
semidefinite, with A = H * Q.
Of course, H being the non-negative semidefinite square root of A*A', it
is unique, but the uniqueness of Q is not guaranteed even if A is
invertible (you can reverse every simple eigenvector, and choose different
orthonormal bases for multiple eigenvalues).
Hope it helps, ZVK(Slavek).
==============================================================================
From: spellucci@mathematik.tu-darmstadt.de (Peter Spellucci)
Subject: Re: Real Polar Decomposition
Date: 4 Feb 1999 17:53:06 GMT
Newsgroups: sci.math.num-analysis
In article <36B84E18.532B09A9@e-technik.uni-ulm.de>,
Bernd Tibken writes:
|> Hello all
|>
|> I would like to have references for the real polar decomposition which
|> states that every real n*n matrix A has a representation as A=H*U where
|> H is symmetric and positive semidefinite and U is orthogonal.
snip
|> Unfortunately I need the result for real matrices.
see 16.7 in
Zurmuehl: "matrizen" 4. Aufl. satz 9. If you look at the proof then
you can see that in the real case AA* is real and A*A real, hence their
eigenvectors can be chosen real and you are done.
hope this helps
peter