ALMOST EVERY UNIT MATRIX IS A ULU

We call an n x n matrix a shear if it is triangular with all 1's on th e diagonal, and a unit matrix if it has unit determinant. Earlier we h ad shown that, for n = 3,every orthogonal matrix (except for degenerat e cases when one of the Euler angles equals pi) can be written in the form U0LU1, where the U are upper shears and L is a lower shear. Then Strang showed that, for any n, every unit matrix can be written as L0U 0L1U1. Here, we show that every unit matrix (except for a subset of me asure zero) can be decomposed into the product of just three shears, U 0LU1, and we present a canonical form for this decomposition. On the r esidual subset, such a decomposition is still possible (up to a sign) if one is allowed to suitably prepermute the rows of the matrix. (C) E lsevier Science Inc., 1997.