Full rank factorization in echelon form of totally nonpositive (negative) rectangular matrices

Cantó, Rafael; Ricarte, Beatriz; Urbano, Ana M.

doi:10.1016/j.laa.2009.07.020

Cited by 11 publications

(15 citation statements)

References 12 publications

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“…In [5] the authors prove that if A = (a ij ) ∈ R n×m is a t.n.p. matrix with a 11 < 0 and rank(A) = r ≤ min{n, m} then, it admits a full rank decomposition in echelon form A = LDU , where L ∈ R n×r is a unit lower echelon TP matrix,…”

Section: Rectangular Tnp Matricesmentioning

confidence: 99%

On the characterization of totally nonpositive matrices

Cantó

Peláez²,

Urbano

2016

SeMA

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A nonpositive real matrix A = (a ij ) 1≤i,j≤n is said to be totally nonpositive (negative) if all its minors are nonpositive (negative) and it is abbreviated as t.n.p. (t.n.). In this work a bidiagonal factorization of a nonsingular t.n.p. matrix A is computed and it is stored in an matrix represented by BD (t.n.p.) (A) when a 11 < 0 (or BD (zero) (A) when a 11 = 0). As a converse result, an efficient algorithm to know if an matrix BD (t.n.p.) (A) (BD (zero) (A)) is the bidiagonal factorization of a t.n.p. matrix with a 11 < 0 (a 11 = 0) is given. Similar results are obtained for t.n. matrices using the matrix BD (t.n.) (A), and these characterizations are extended to rectangular t.n.p. (t.n.) matrices. Finally, the bidiagonal factorization of the inverse of a nonsingular t.n.p. (t.n.) matrix A is directly obtained from BD (t.n.p.) (A) (BD (t.n.) (A)).

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Section: Rectangular Tnp Matricesmentioning

confidence: 99%

On the characterization of totally nonpositive matrices

Cantó

Peláez²,

Urbano

2016

SeMA

Self Cite

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“…(t.n.) see, for instance, [3,4,6,8,14,15,18]. These matrices can be considered as a generalization of the partially negative matrices, that is, matrices with all its principal minors negative.…”

Section: Introductionmentioning

confidence: 99%

“…matrix by permutation similarity. Then, we have studied in [4] the extension of the same characterization to rectangular t.n.p. matrices, obtaining a full rank LDU factorization in echelon form of this class of matrices and other characterization by means of its thin QR factorization.…”

Section: Introductionmentioning

confidence: 99%

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Full rank factorization in quasi- LDU form of totally nonpositive rectangular matrices

Cantó

Ricarte

Urbano

2014

Linear Algebra and its Applications

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Let A = (a ij ) ∈ R n×m be a totally nonpositive matrix with rank(A) = r ≤ min{n, m} and a 11 = 0. In this paper we obtain a characterization in terms of the full rank factorization in quasi-LDU form, that is, A =LDU wherẽ L ∈ R n×r is a block lower echelon matrix, U ∈ R r×m is a unit upper echelon totally positive matrix and D ∈ R r×r is a diagonal matrix, with rank(L) = rank(U ) = rank(D) = r. We use this quasi-LDU decomposition to construct the quasi-bidiagonal factorization of A. Moreover, some properties about these matrices are studied.

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“…Specifically, in [1,5] it is proven that totally positive (resp., strictly totally positive) matrices, that is matrices with all theirs minors greater than or equal to zero (resp., greater than zero), and totally nonpositive (resp. negative) matrices, that is matrices with all theirs minors less than or equal to zero (resp.…”

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confidence: 99%