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Totally Positive Wronskian Matrices and Symmetric Functions [PDF]
AxiomsThe elements of the bidiagonal decomposition (BD) of a totally positive (TP) collocation matrix can be expressed in terms of symmetric functions of the nodes. Making use of this result, and studying the relation between Wronskian and collocation matrices
Pablo Díaz +2 more
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Sums of totally positive matrices [PDF]
Linear Algebra and Its Applications, 2004 An \(m\times n\) matrix is called totally positive if all of its minors are positive. The authors prove that an arbitrary \(m\times n\) positive matrix can be written as a sum of at most \(\min\{m, n\}\) totally positive matrices. This generalizes the fact that a positive matrix is the sum of two matrices whose all principal minors are positive.
D D Olesky
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Totally positive matrices [PDF]
Linear Algebra and Its Applications, 1987 As a survey paper on totally positive matrices, this article enhances the earlier work of Gantmacher and Krein, and Karlin. There are seven short sections, each complete with definitions, theorems, proofs and references on: determinantal identities in light of tensor products and Schur complements; criteria for total positivity using sign-regularity ...
Ando, T.
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M-matrices whose inverses are totally positive [PDF]
Linear Algebra and Its Applications, 1995 A real invertible \(n \times n\) matrix with nonpositive off-diagonal elements is an \(M\)-matrix provided \(Ax \geq 0\) implies \(x \geq 0\) for all \(x \in {\mathcal F}^n\). The author shows that if \(A\) is such a matrix, then \(A^{- 1}\) is totally positive (i.e., all minors are nonnegative) if and only if \(A\) is a tridiagonal matrix.
J M Pena
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Totally positive matrices and totally positive hypergraphs [PDF]
Linear Algebra and its Applications, 2001 This paper characterizes (0,1)-matrices which are totally positive, that is, all their minors are totally positive. First the case of \(1\times 1\) and \(2\times 2\) minors is characterized in terms of interval hypergraphs and then the general case is characterized in terms of a chain of cliques \(C_i\), \(C_i\cap C_j = \emptyset \Leftrightarrow |i-j ...
Kubicki, Grzegorz +2 more
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Tropical totally positive matrices [PDF]
Journal of Algebra, 2018 The first author has been partially supported by the PGMO Program of FMJH and EDF, and by the MALTHY Project of the ANR Program.
Gaubert, Stéphane, Niv, Adi
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On Factorizations of Totally Positive Matrices [PDF]
, 1996 Different approaches to the decomposition of a nonsingular totally positive matrix as a product of bidiagonal matrices are studied. Special attention is paid to the interpretation of the factorization in terms of the Neville elimination process of the matrix and in terms of corner cutting algorithms of Computer Aided Geometric Design.
Mariano Gasca, Juan M. Peña
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On totally positive matrices and geometric incidences
Journal of Combinatorial Theory - Series A, 2014 11 ...
Miriam Farber +2 more
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Generalized totally positive matrices [PDF]
Linear Algebra and its Applications, 2000 A matrix over a ring with identity and a positive part is called generalized totally positive (GTP) if the Schur complements are positive in all nested sequences of so-called relevant submatrices, i.e. ones having either the first \(k\) rows and \(k\) consecutive columns, or \(k\) consecutive rows and the first \(k\) columns.
Fiedler, Miroslav, Markham, Thomas L.
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Intervals of almost totally positive matrices [PDF]
Linear Algebra and its Applications, 2003 The author shows that a stronger form of the total nonnegativity is preserved on matrix intervals with respect to the chequerboard partial ordering on the real nonsingular matrices. The analogous question for totally nonnegative matrices seems to remain open since 1996.
Garloff, Jürgen
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