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Every unit matrix is a LULU. (English) Zbl 0918.15004
This paper is devoted to prove that every unit matrix \(A\) (i.e. every matrix with determinant one) has a factorization of the form \(A=L_0 U_0 L_1 U_1\), where the four matrices are triangular with ones on their diagonal (\(L_i\) are lower triangular, and \(U_i\) are upper triangular). The paper should be read in connection to T. Toffoli’s paper [ibid. 259, 31-38 (1997; Zbl 0893.15004)] where a related result, motivated with examples from computer graphics, is stated.
In the paper under review, the author proves the existence of a factorization as a product of four triangular matrices for every unit matrix (not almost every). In addition, it should be mentioned that the problem of finding an efficient method to construct the factorization (“a minimal adjustment” in the author’s words) remains open.

MSC:
15A23 Factorization of matrices
15B57 Hermitian, skew-Hermitian, and related matrices
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