Perhaps "unique" means unique only once you choose whether you want the symmetric matrix on the left or the right? At 03:26 PM 9/8/2010, Fred lunnon 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