Fred, Here might be the source of confusion. The matrix M has a singular value decomposition M = U D V where U, V are orthogonal and D is a diagonal matrix with the nonnegative values in the upper left hand and sorted in descending order. Such a D is unique, however, if diagonal values of D are equal, the U and V are not unique (since you can just interchange them and appropriately modify U and V). The relation between this and the polar decomposition is that a symmetric matrix S can be written in the form U D U^-1, which is almost unique (the same problem arises when two diagonal values of D are equal or opposite sign. So take the above singular value decomposition and write M = (U D U^-1) (U V). Note that the first factor is symmetric (and all symmetric factors arise this way) and the second factor orthogonal. Victor On Wed, Sep 8, 2010 at 6:26 PM, Fred lunnon <fred.lunnon@gmail.com> wrote:
In numerous recent publications, it is claimed that the "polar decomposition" of a (real) matrix M = Q S as the product of orthonormal ("orthogonal") and symmetric factors is unique, at any rate provided M is nonsingular.
See for example http://en.wikipedia.org/wiki/Polar_decomposition or the very readable (and available for free download) Ken Shoemake, Tom Duff "Matrix Animation and Polar Decomposition"
As it stands, the claim is obviously false. A trivial example is M = [[0,1],[1,0]] which factorises as M I or as I M.
There is no mention of uniqueness on page 6 of Gantmacher "Theory of Matrices" vol 2 though he does point out that the factors commute if M commutes with its transpose, as in the counterexample above.
The claim of uniqueness may be possibly result from confusion connected with the Higham 1986 reference in Shoemake, where there is an algorithm giving Q such that Q - M has minimal 2-norm.
Can anyone cast light on this? Is the decomposition unique under the minimality constraint? Are there less onerous constraints guaranteeing uniqueness (and minimality)?
Have I simply misunderstood something, or is this another rhinoceros' pancreas?
Fred Lunnon
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