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On the necessity and sufficiency of \(PLUS\) factorizations. (English) Zbl 1071.15012
A \(PLUS\) factorization for an arbitrary nonsingular \(n\times n\) matrix \(A\) has the form \(A=PLUS\), where \(P\) is a permutation matrix, \(L\) is a unit lower triangular matrix, \(U\) is an upper triangular matrix whose diagonal entries are prescribed as long as the determinant is equal to that of \(A\) up to a possible sign adjustment, and \(S\) is a unit lower triangular matrix of which all but \(n-1\) off-diagonal entries are zeros and the positions of those \(n-1\) entries are also flexibly customizable.
The authors show that the necessary condition for the existence of a \(PLUS\) factorization of a matrix \(A\) as given by P. Hao [ibid. 382, 135–154 (2004; Zbl 1050.15012)] is not sufficient and they find a sufficient condition for such a factorization.

15A23 Factorization of matrices
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[1] Chen, B.; Kaufman, A., 3D volume rotation using shear transformations, Gmip, 62, 308-322, (2000)
[2] Daubechies, I.; Sweldens, W., Factoring wavelet transforms into lifting steps, J. Fourier anal. appl., 4, 3, 247-269, (1998) · Zbl 0913.42027
[3] Hao, P.; Shi, Q., Matrix factorizations for reversible integer mapping, IEEE trans. signal process., 49, 10, 2314-2324, (2001) · Zbl 1369.94025
[4] Hao, P., Customizable triangular factorizations of matrices, Linear algebra appl., 282, 135-154, (2004) · Zbl 1050.15012
[5] Nagarajan, K.R.; Devasahayam, M.P.; Soundararajan, T., Products of three triangular matrices, Linear algebra appl., 292, 61-71, (1999) · Zbl 0933.15021
[6] Y. She, Matrix factorizations for efficient implementation of linear transforms, Master Thesis, Peking University, July 2003 (in Chinese)
[7] She, Y.; Hao, P., Block TERM factorizations for uniform block matrices, Sci. China (ser. F), 47, 4, 421-436, (2004) · Zbl 1186.15012
[8] Strang, G., Every unit matrix is a LULU, Linear algebra appl., 265, 165-172, (1997) · Zbl 0918.15004
[9] Toffoli, T., Almost every unit matrix is a ULU, Linear algebra appl., 259, 31-38, (1997) · Zbl 0893.15004
[10] Vaserstein, L.N.; Wheland, E., Commutators and companion matrices over rings of stable rank 1, Linear algebra appl., 142, 263-277, (1990) · Zbl 0713.15003
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