Products, coproducts and singular value decomposition

Details Products, coproducts and singular value decomposition Products, coproducts and singular value decomposition

2004-02-28

arxiv.org/abs/math-ph/0403001v1

Int. J. Theor. Phys. Vol 45, No 9, 2006: 1731-1755

Physics

Mathematical Physics

17 pages, three eps-figures

Scientific paper

10.1007/s10773-006-9111-6

Products and coproducts may be recognized as morphisms in a monoidal tensor category of vector spaces. To gain invariant data of these morphisms, we can use singular value decomposition which attaches singular values, ie generalized eigenvalues, to these maps. We show, for the case of Grassmann and Clifford products, that twist maps significantly alter these data reducing degeneracies. Since non group like coproducts give rise to non classical behavior of the algebra of functions, ie make them noncommutative, we hope to be able to learn more about such geometries. Remarkably the coproduct for positive singular values of eigenvectors in $A$ yields directly corresponding eigenvectors in A\otimes A.

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