Cantó, Begoña; Cantó, Rafael; Urbano, Ana María Jordan structures of irreducible totally nonnegative matrices with a prescribed sequence of the first p-indices. (English) Zbl 1486.15017 Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No. 2, Paper No. 84, 27 p. (2022). Summary: Let \(A \in\mathbb{R}^{n \times n}\) be an irreducible totally nonnegative matrix with rank \(r\) and principal rank \(p\), that is, \(A\) is irreducible with all minors nonnegative, \(r\) is the size of the largest invertible square submatrix of \(A\) and \(p\) is the size of its largest invertible principal submatrix. We consider the sequence \(\{1, i_2, \dots, i_p\}\) of the first \(p\)-indices of \(A\) as the first initial row and column indices of a \(p \times p\) invertible principal submatrix of \(A\). A triple \((n, r, p)\) is called \((1, i_2, \dots, i_p)\)-realizable if there exists an irreducible totally nonnegative matrix \(A \in\mathbb{R}^{n \times n}\) with rank \(r\), principal rank \(p\), and \(\{1, i_2, \dots, i_p\}\) is the sequence of its first \(p\)-indices. In this work we study the Jordan structures corresponding to the zero eigenvalue of irreducible totally nonnegative matrices associated with a triple \((n, r, p)\) \((1, i_2, \dots, i_p)\)-realizable. MSC: 15A20 Diagonalization, Jordan forms 15A21 Canonical forms, reductions, classification 15A03 Vector spaces, linear dependence, rank, lineability 15A29 Inverse problems in linear algebra Keywords:totally nonnegative matrix; irreducible matrix; totally nonpositive matrix; triple realizable; Jordan canonical form; linear algebra PDFBibTeX XMLCite \textit{B. Cantó} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 116, No. 2, Paper No. 84, 27 p. (2022; Zbl 1486.15017) Full Text: DOI References: [1] Ando, T., Totally positive matrices, Linear Algebra Appl., 90, 165-219 (1987) · Zbl 0613.15014 · doi:10.1016/0024-3795(87)90313-2 [2] Bru, R.; Cantó, R.; Urbano, AM, Eigenstructure of rank one updated matrices, Linear Algebra Appl., 485, 372-391 (2015) · Zbl 1323.15001 · doi:10.1016/j.laa.2015.07.036 [3] Brualdi, RA; Ryser, HJ, Combinatorial Matrix Theory (1991), Canada: Cambridge University Press, Canada · Zbl 1286.05001 · doi:10.1017/CBO9781107325708 [4] Cantó, R.; Urbano, AM, On the maximum rank of totally nonnegative matrices, Linear Algebra Appl., 551, 125-146 (2018) · Zbl 1415.15001 · doi:10.1016/j.laa.2018.03.045 [5] Cantó, B.; Cantó, R.; Urbano, AM, All Jordan canonical forms of irreducible totally nonnegative matrices, Linear Multilinear Algebra, 69, 2389-2409 (2021) · Zbl 1478.15016 · doi:10.1080/03081087.2019.1676691 [6] Cantó, B.; Cantó, R.; Urbano, AM, Irreducible totally nonnegative matrices with a precribed Jordan structure, Linear Algebra Appl., 609, 129-151 (2021) · Zbl 1457.15032 · doi:10.1016/j.laa.2020.09.001 [7] Cantó, B.; Cantó, R.; Urbano, AM, On totally nonpositive matrices associated with a triple negatively realizable, RACSAM Rev R Acad A, 134, 1-25 (2021) · Zbl 1472.15001 · doi:10.1007/s13398-021-01073-9 [8] Fallat, SM; Gekhtman, MI, Jordan structure of totally nonnegative matrices, Can. J. Math., 57, 82-98 (2005) · Zbl 1072.15029 · doi:10.4153/CJM-2005-004-0 [9] Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton Ser. Appl. Math. (2011) · Zbl 1390.15001 [10] Fallat, SM; Gekhtman, MI; Johnson, CR, Spectral structures of irreducible totally nonnegative matrices, SIAM J. Matrix Anal. Appl., 22, 627-645 (2000) · Zbl 0970.15013 · doi:10.1137/S0895479800367014 [11] Gantmacher, F.R., Krein, M.G.: Ostsillyatsionye Matritsy i Yadra i Malye Kolebaniya Mekhanicheskikh Sistem, Gosudarstvenoe Izdatel’stvo, Moskva-Leningrad, 1950. (Englishtransl. as “Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems”), USAEC (1961) · Zbl 0041.35502 [12] Gupta, SB; Gandhi, CP, Discrete Structures (2010), New Delhi: University Science Press, New Delhi [13] Pinkus, A., Totally Positive Matrices (2010), New York: Cambridge University Press, New York · Zbl 1185.15028 [14] Shapiro, H., The Weyr characteristic, Am. Math. Mon., 106, 919-929 (1999) · Zbl 0981.15008 · doi:10.1080/00029890.1999.12005141 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.