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