Almost every unit matrix is a ULU Original Research Article

We call an n × n matrix a shear if it is triangular with all lʹs 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 π) 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 L0U0L1U1. Here, we show that every unit matrix (except for a subset of measure zero) can be decomposed into the product of just three shears, U0LU1, 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.