Book contents
- Frontmatter
- Contents
- Foreword
- 1 Basic properties of totally positive and strictly totally positive matrices
- 2 Criteria for total positivity and strict total positivity
- 3 Variation diminishing
- 4 Examples
- 5 Eigenvalues and eigenvectors
- 6 Factorizations of totally positive matrices
- Afterword
- References
- Author index
- Subject index
- References
References
Published online by Cambridge University Press: 05 May 2010
Book contents
- Frontmatter
- Contents
- Foreword
- 1 Basic properties of totally positive and strictly totally positive matrices
- 2 Criteria for total positivity and strict total positivity
- 3 Variation diminishing
- 4 Examples
- 5 Eigenvalues and eigenvectors
- 6 Factorizations of totally positive matrices
- Afterword
- References
- Author index
- Subject index
- References
Summary
A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
- Type
- Chapter
- Information
- Totally Positive Matrices , pp. 174 - 179Publisher: Cambridge University PressPrint publication year: 2009
References
, and [1952], On the generating functions of totally positive sequences I, J. d'Anal. Math. 2, 93–103.CrossRefGoogle Scholar
[1970], On the total nonnegativity of the Hurwitz matrix, SIAM J. Appl. Math. 18, 407–414.CrossRefGoogle Scholar
, and [1996], Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122, 49–149.CrossRefGoogle Scholar
and [2008], On generators of bounded ratios of minors for totally positive matrices, Lin. Alg. and Appl. 428, 1664–1684.CrossRefGoogle Scholar
[1982], The inverse of totally positive bi-infinite band matrices, Trans. Amer. Math. Soc. 274, 45–58.CrossRefGoogle Scholar
[1988], I. J. Schoenberg: Selected Papers, 2 Volumes, Birkhäuser, Basel.
and [1977], Backward error analysis for totally positive linear systems, Numer. Math. 27, 485–490.CrossRefGoogle Scholar
and [1982], The approximation of a totally positive band matrix by a strictly banded totally positive one, Lin. Alg. and Appl. 42, 81–98.CrossRefGoogle Scholar
[1989], Unimodal, log-concave, and P?olya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 413.Google Scholar
[1995], Combinatorics and total positivity, J. Comb. Theory, Series A 71, 175–218.CrossRefGoogle Scholar
[1996], The applications of total positivity to combinatorics and conversely, inTotal Positivity and its Applications eds., and , Kluwer Acad. Publ., Dordrecht, 451–473.CrossRefGoogle Scholar
, and [1981], Variation diminishing transformations: a direct approach to total positivity and its statistical applications, J. Amer. Stat. Assoc. 76, 824–832.CrossRefGoogle Scholar
, and [1983], Determinantal identities: Gauss, Schur, Cauchy, Dylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Lin. Alg. and Appl. 52/53, 769–791.CrossRefGoogle Scholar
[1990], A variational description of the spectra of totally positive matrices, and extremal problems of approximation theory, Mat. Zametki 47, 39–46; English transl. in Math. Notes 47(1990), 26–31.Google Scholar
and [1983], Total positivity of mean values and hypergeometric functions, SIAM J. Math. Anal. 14, 389–395.CrossRefGoogle Scholar
[1968], On some determinantal inequalities, Proc. Amer. Math. Soc. 19, 462–466.CrossRefGoogle Scholar
, and [1995], A generalization of the variation diminishing property, Adv. Comp. Math. 3, 375–394.CrossRefGoogle Scholar
, , and [1981], A factorization theorem for banded matrices, Lin. Alg. and Appl. 39, 229–245.CrossRefGoogle Scholar
, and [2001], The Hadamard core of the totally nonnegative matrices, Lin. Alg. and Appl. 328, 203–222.CrossRefGoogle Scholar
and [1998], A sufficient condition for strict total positivity of a matrix, Linear and Multilinear Alg. 45, 19–34.CrossRefGoogle Scholar
[1973], The LU-factorization of totally positive matrices, Lin. Alg. and Appl. 7, 83–92.CrossRefGoogle Scholar
[1976], Some properties of totally positive matrices, Lin. Alg. and Appl. 15, 1–25.CrossRefGoogle Scholar
, , and [1986], On factorization of bi-infinite totally positive block Toeplitz matrices, Rocky Mountain J. Math. 16, 335–364.CrossRefGoogle Scholar
and [2005], The accurate and efficient solution of a totally positive generalized Vandermonde linear system, SIAM J. Matrix. Anal. Appl. 27, 142–152.CrossRefGoogle Scholar
and [2005], Almost strict total positivity and a class of Hurwitz polynomials, J. Approx. Theory 132, 212–223.CrossRefGoogle Scholar
[1952], On the generating functions of totally positive sequences II, J. d'Anal. Math. 2, 104–109.CrossRefGoogle Scholar
[1953], Proof of a conjecture of Schoenberg on the generating functions of totally positive sequences, Canadian J. Math. 5, 86–94.CrossRefGoogle Scholar
and [2002], Non-linear eigenvalue-eigenvector problems for STP matrices, Proc. Royal Society Edinburgh: Section A 132, 1307–1331.Google Scholar
[1996], The eigenvalue distribution of oscillatory and strictly sign-regular matrices, Lin. Alg. and Appl. 246, 17–21.CrossRefGoogle Scholar
, and [2000], Spectral structures of irreducible totally nonnegative matrices, SIAM J. Matrix Anal. Appl. 22, 627–645.CrossRefGoogle Scholar
, and [2003], Multiplicative principal-minor inequalities for totally nonnegative matrices, Adv. Applied Math. 30, 442–470.CrossRefGoogle Scholar
[1967], Subadditive functions on a distributive lattice and an extension of Szasz's inequality, J. Math. Anal. Appl. 18, 262–268.CrossRefGoogle Scholar
[1968], An inequality for subadditive functions on a distributive lattice with application to determinantal inequalities, Lin. Alg. and Appl. 1, 33–38.CrossRefGoogle Scholar
and [1999], Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12, 335–380.CrossRefGoogle Scholar
and [2000], Total positivity: tests and parametrizations, Math. Intell. 22, 23–33.CrossRefGoogle Scholar
[1985], Weak interlacing properties of totally positive matrices, Lin. Alg. and Appl. 71, 95–100.CrossRefGoogle Scholar
[1936],Sur les noyaux de Kellogg non symétriques, Comptes Rendus (Doklady) de l'Academie des Sciences de l'URSS 1 (10), 3–5.Google Scholar
[1953], The Theory of Matrices, Gostekhizdat, Moscow- Leningrad; English transl. as Matrix Theory, Chelsea, New York, 2 vols., 1959.Google Scholar
[1965], Obituary, in Uspekhi Mat. Nauk 20, 149–158; English transl. as Russian Math. Surveys, 20(1965), 143–151.
and [1937], Sur les matrices compl`etement non négatives et oscillatoires, Compositio Math. 4, 445–476.Google Scholar
and [1941], Oscillation Matrices and Small Oscillations of Mechanical Systems (Russian), Gostekhizdat, Moscow-Leningrad.
and [1950], Ostsillyatsionye Matritsy i Yadra i Malye Kolebaniya Mekhanicheskikh Sistem, Gosudarstvenoe Izdatel'stvo, Moskva-Leningrad, 1950; German transl. as Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme, Akademie Verlag, Berlin, 1960; English transl. as Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems, USAEC, 1961, and also a revised English edition from AMS Chelsea Publ., 2002.
[1982], Criteria for sign regularity of sets of matrices, Lin. Alg. and Appl. 44, 153–160.CrossRefGoogle Scholar
[2003], Intervals of almost totally positive matrices, Lin. Alg. and Appl. 363, 103–108.CrossRefGoogle Scholar
and [1996a], Hadamard products of stable polynomials are stable, J. Math. Anal. Appl. 202, 797–809.CrossRefGoogle Scholar
and [1996b], Preservation of total nonnegativity under the Hadamard products and related topics, in Total Positivity and its Applications eds., and , Kluwer Acad. Publ., Dordrecht, 97–102.CrossRefGoogle Scholar
and [1996], Total Positivity and its Applications, eds., Kluwer Acad. Publ., Dordrecht.CrossRefGoogle Scholar
, and [1992], Almost strictly totally positive matrices, Numerical Algorithms 2, 225–236.CrossRefGoogle Scholar
and [1992], Total positivity and Neville elimination, Lin. Alg. and Appl. 165, 25–44.CrossRefGoogle Scholar
and [1993], Total positivity, QR factorization, and Neville elimination, SIAM J. Matrix Anal. Appl. 14, 1132–1140.CrossRefGoogle Scholar
and [1995], On the characterization of almost strictly totally positive matrices, Adv. Comp. Math. 3, 239–250.CrossRefGoogle Scholar
and [1996], On factorizations of totally positive matrices, in Total Positivity and its Applications eds., and , Kluwer Acad. Publ., Dordrecht, 109–130.CrossRefGoogle Scholar
and [2006], Characterizations and decompositions of almost strictly totally positive matrices, SIAM J. Matrix Anal. 28, 1–8.CrossRefGoogle Scholar
[1998], Total positivity and the QR algorithm, Lin. Alg. and Appl. 271, 257–272.CrossRefGoogle Scholar
[1989], Mathematical Tales, in The Gohberg Anniversary Collection, Eds. , , , , pp. 17–56, Operator Theory: Advances and Applications, Vol. 40, Birkhäuser Verlag, Basel.CrossRefGoogle Scholar
[1995], Total positivity and the shape of curves, in Total Positivity and its Applications eds., and , Kluwer Acad. Publ., Dordrecht, 157–186.Google Scholar
and [2004], Total positivity and refinable functions with general dilation, Appl. Comput. Harmon. Anal. 16, 69–89.CrossRefGoogle Scholar
and [1995], Total positivity, finite reflection groups, and a formula of Harish-Chandra, J. Approx. Theory 82, 60–87.CrossRefGoogle Scholar
[2003], Hermite-Biehler, Routh-Hurwitz and total positivity, Lin. Alg. and Appl. 372, 105–110.CrossRefGoogle Scholar
[1964], The existence of eigenvalues for integral operators, Trans. Amer. Math. Soc. 113, 1–17.CrossRefGoogle Scholar
[1965], Oscillation properties of eigenvectors of strictly totally positive matrices, J. D'Analyse Math. 14, 247–266.CrossRefGoogle Scholar
[1972], Some extremal problems for eigenvalues of certain matrix and integral operators, Adv. in Math. 9, 93–136.CrossRefGoogle Scholar
[2002], Interdisciplinary meandering in science. 50th anniversary issue of Operations Research. Oper. Res. 50, 114–121.CrossRefGoogle Scholar
and [1974], Oscillation properties of generalized characteristic polynomials for totally positive and positive definite matrices, Lin. Alg. and Appl. 8, 281–312.CrossRefGoogle Scholar
and [1966], Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience Publishers, John Wiley, New York.Google Scholar
and [2006], On sufficient conditions for the total positivity and for the multiple positivity of matrices, Lin. Alg. and Appl. 416, 1083–1097.CrossRefGoogle Scholar
[1918], Orthogonal function sets arising from integral equations, Amer. J. Math. 40, 145–154.CrossRefGoogle Scholar
[1982], A Hurwitz matrix is totally positive, SIAM J. Math. Anal. 13, 331–341.CrossRefGoogle Scholar
[2005], Accurate eigenvalues and SVDs of totally nonnegative matrices, SIAM J. Matrix. Anal. Appl. 27, 1–23.CrossRefGoogle Scholar
[2007], Accurate computations with totally nonnegative matrices, SIAM J. Matrix. Anal. Appl. 29, 731–751.CrossRefGoogle Scholar
[1950], The theory of nonnegative and oscillating matrices (Russian), Ukrain. Mat. Z. 2, 94–101; English transl. in Amer. Math. Soc. Transl., Series 2 27(1963), 1–8.Google Scholar
[1955], Some sufficient conditions for reality and simplicity of the spectrum of a matrix, Mat. Sb. N.S. 36, 163–168; English transl. in Amer. Math. Soc. Transl., Series 2 27(1963), 35–42.Google Scholar
and [1977], The Markov Moment Problem and Extremal Problems, Transl. Math. Monographs, Vol. 50, Amer. Math. Society, Providence.CrossRef
[1992], A sufficient condition for all the roots of a polynomial to be real, Amer. Math. Monthly 99, 259–263.CrossRefGoogle Scholar
[1994], Total positivity in reductive groups, in Lie Theory and Geometry, Progress in Mathematics, 123, Birkhäuser, Boston, 531–568.Google Scholar
[1895], Note sur les équations algébriques dont toutes les racines sont réelles, Journal de Mathématiques Spéciales 4, 7–10.Google Scholar
[1966], Geometry of Polynomials, Math. Surveys, Vol. 3 (2nd edition), Amer. Math. Society, Providence.
[1970], A semi-group of totally nonnegative matrices, Lin. Alg. and Appl. 3, 157–164.CrossRefGoogle Scholar
and [1979], Inequalities: Theory of Majorization and its Applications, Academic Press, New York.Google Scholar
[1973], Ein Kriterium für den Nachweis der Totalnichtnegativität von Bandmatrizen, Lin. Alg. and Appl. 7, 163–171.CrossRef
and [1991], Descartes systems from corner cutting, Constr. Approx. 7, 161–194.CrossRefGoogle Scholar
[1936], Beiträge zur Theorie der linearen Ungleichungen, Doctoral Dissertation, Basel, 1933. Azriel Press, Jerusalem.Google Scholar
and [1985], A generalization of Sylvester's identity on determinants and some applications, Lin. Alg. and Appl. 66, 221–234.CrossRefGoogle Scholar
[1985a], Some extremal problems for strictly totally positive matrices, Lin. Alg. and Appl. 64, 141–156.CrossRefGoogle Scholar
[1996], Spectral properties of totally positive kernels and matrices, in Total Positivity and its Applications eds., and , Kluwer Acad. Publ., Dordrecht, 477–511.CrossRefGoogle Scholar
[1998], An interlacing property of eigenvalues of strictly totally positive matrices, Lin. Alg. and Appl. 279, 201–206.CrossRefGoogle Scholar
[2008], Zero minors of totally positive matrices, Electronic J. Linear Algebra 17, 532–542.CrossRefGoogle Scholar
[1997], Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Comb. Theory, Ser. A 77, 279–303.CrossRefGoogle Scholar
[1930], Über variationsvermindernde lineare Transformationen, Math. Z. 32, 321–328.CrossRefGoogle Scholar
and [1951], A theorem on polygons in n dimensions with application to variation-diminishing and cyclic variation-diminishing linear transformations, Compositio Math. 9, 141–160.Google Scholar
and [1995], On the boundary of totally positive upper triangular matrices, Lin. Alg. and Appl. 231, 105–109.CrossRefGoogle Scholar
[2004], Inequalities in products of minors of totally nonnegative matrices, J. Alg. Comb. 20, 195–211.CrossRefGoogle Scholar
[1983], Truncation and factorization of bi-infinite matrices, in Approximation Theory, IV (College Station, Tex., 1983), 257–289, Academic Press, New York.Google Scholar
[1894–95], Recherches sur les fractions continues, Annales Fac. Sciences Toulouse 8, 1–122; 9, 1–47.CrossRefGoogle Scholar
[1992], Total positivity of Hadamard products, J. Math. Anal. Appl. 163, 459–483.CrossRefGoogle Scholar
and [2005], Polynomials with real zeros and Pólya frequency sequences, J. Comb. Theory, Ser. A 109, 63–74.CrossRefGoogle Scholar
[1952], A reduction theorem for totally positive matrices, J. d'Anal. Math. 2, 88–92.CrossRefGoogle Scholar