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Almost every unit matrix is a ULU

✍ Scribed by Tommaso Toffoli

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 443 KB
Volume: 259
Category: Article
ISSN: 0024-3795
DOI: 10.1016/s0024-3795(96)00240-6

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✦ Synopsis

We call an n X n matrix a shear if it is triangular with all l's on the diagonal, and a unit matrix if it has unit determinant. Earlier we had shown that, for n = 3, every orthogonal matrix (except for degenerate cases when one of the Euler angles equals rr) can be written in the form U,,LU,, where the U are upper shears and Z. is a lower shear. Then Strang showed that, for any n, every unit matrix can be written as L,U,L,U,.

Here, we show that every unit matrix (except for a subset of measure zero) can be decomposed into the product of just three shears, U,LU,, and we present a canonical form for this decomposition.

On the residual subset, such a decomposition is still possible (up to a sign) if one is allowed to suitably prepermute the rows of the matrix.

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