Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T15:03:13.149Z Has data issue: false hasContentIssue false

References

Published online by Cambridge University Press:  05 May 2010

Allan Pinkus
Allan Pinkus
Affiliation:
Technion - Israel Institute of Technology, Haifa
Get access

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.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Totally Positive Matrices , pp. 174 - 179
Publisher: Cambridge University Press
Print publication year: 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Aissen, M., Schoenberg, I. J. and Whitney, A. M. [1952], On the generating functions of totally positive sequences I, J. d'Anal. Math. 2, 93–103.CrossRefGoogle Scholar
Ando, T. [1987], Totally positive matrices, Lin. Alg. and Appl. 90, 165–219.CrossRefGoogle Scholar
Askey, R. and Boor, C. [1990], In memoriam : Schoenberg, I. J. (1903–1990), J. Approx. Theory 63, 1–2.CrossRef
Asner, B. A. Jr. [1970], On the total nonnegativity of the Hurwitz matrix, SIAM J. Appl. Math. 18, 407–414.CrossRefGoogle Scholar
Beckenbach, E. F. and Bellman, R. [1961], Inequalities, Springer–Verlag, New York.CrossRefGoogle Scholar
Berenstein, A., Fomin, S. and Zelevinsky, A. [1996], Parametrizations of canonical bases and totally positive matrices, Adv. Math. 122, 49–149.CrossRefGoogle Scholar
Boocher, A. and Froehle, B. [2008], On generators of bounded ratios of minors for totally positive matrices, Lin. Alg. and Appl. 428, 1664–1684.CrossRefGoogle Scholar
Boor, C. [1982], The inverse of totally positive bi-infinite band matrices, Trans. Amer. Math. Soc. 274, 45–58.CrossRefGoogle Scholar
Boor, C. [1988], I. J. Schoenberg: Selected Papers, 2 Volumes, Birkhäuser, Basel.
Boor, C. and Pinkus, A. [1977], Backward error analysis for totally positive linear systems, Numer. Math. 27, 485–490.CrossRefGoogle Scholar
Boor, C. and Pinkus, A. [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
Brenti, F. [1989], Unimodal, log-concave, and P?olya frequency sequences in combinatorics, Mem. Amer. Math. Soc. 413.Google Scholar
Brenti, F. [1995], Combinatorics and total positivity, J. Comb. Theory, Series A 71, 175–218.CrossRefGoogle Scholar
Brenti, F. [1996], The applications of total positivity to combinatorics and conversely, inTotal Positivity and its Applications eds., Micchelli, C. A. and Gasca, M., Kluwer Acad. Publ., Dordrecht, 451–473.CrossRefGoogle Scholar
Brown, L. D., Johnstone, I. M. and MacGibbon, K. B. [1981], Variation diminishing transformations: a direct approach to total positivity and its statistical applications, J. Amer. Stat. Assoc. 76, 824–832.CrossRefGoogle Scholar
Brualdi, R. A., and Schneider, H. [1983], Determinantal identities: Gauss, Schur, Cauchy, Dylvester, Kronecker, Jacobi, Binet, Laplace, Muir, and Cayley, Lin. Alg. and Appl. 52/53, 769–791.CrossRefGoogle Scholar
Buslaev, A. P. [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
Carlson, B. C. and Gustafson, J. L. [1983], Total positivity of mean values and hypergeometric functions, SIAM J. Math. Anal. 14, 389–395.CrossRefGoogle Scholar
Carlson, D. [1968], On some determinantal inequalities, Proc. Amer. Math. Soc. 19, 462–466.CrossRefGoogle Scholar
Carnicer, J. M., Goodman, T. N. T. and , J. M. [1995], A generalization of the variation diminishing property, Adv. Comp. Math. 3, 375–394.CrossRefGoogle Scholar
Cavaretta, A. S. Jr., Dahmen, W. A. and Micchelli, C. A. [1991], Stationary subdivision, Mem. Amer. Math. Soc. 93.Google Scholar
Cavaretta, A. S. Jr., Dahmen, W. A., Micchelli, C. A. and Smith, P. W. [1981], A factorization theorem for banded matrices, Lin. Alg. and Appl. 39, 229–245.CrossRefGoogle Scholar
Crans, A. S., Fallat, S. M. and Johnson, C. R. [2001], The Hadamard core of the totally nonnegative matrices, Lin. Alg. and Appl. 328, 203–222.CrossRefGoogle Scholar
Craven, T. and Csordas, G. [1998], A sufficient condition for strict total positivity of a matrix, Linear and Multilinear Alg. 45, 19–34.CrossRefGoogle Scholar
Cryer, C. [1973], The LU-factorization of totally positive matrices, Lin. Alg. and Appl. 7, 83–92.CrossRefGoogle Scholar
Cryer, C. [1976], Some properties of totally positive matrices, Lin. Alg. and Appl. 15, 1–25.CrossRefGoogle Scholar
Dahmen, W., Micchelli, C. A., and Smith, P. W. [1986], On factorization of bi-infinite totally positive block Toeplitz matrices, Rocky Mountain J. Math. 16, 335–364.CrossRefGoogle Scholar
Demmel, J. and Koev, P. [2005], The accurate and efficient solution of a totally positive generalized Vandermonde linear system, SIAM J. Matrix. Anal. Appl. 27, 142–152.CrossRefGoogle Scholar
Dimitrov, D. K. and Peña, J. M. [2005], Almost strict total positivity and a class of Hurwitz polynomials, J. Approx. Theory 132, 212–223.CrossRefGoogle Scholar
Edrei, A. [1952], On the generating functions of totally positive sequences II, J. d'Anal. Math. 2, 104–109.CrossRefGoogle Scholar
Edrei, A. [1953], Proof of a conjecture of Schoenberg on the generating functions of totally positive sequences, Canadian J. Math. 5, 86–94.CrossRefGoogle Scholar
Elias, U. and Pinkus, A. [2002], Non-linear eigenvalue-eigenvector problems for STP matrices, Proc. Royal Society Edinburgh: Section A 132, 1307–1331.Google Scholar
Eveson, S. P. [1996], The eigenvalue distribution of oscillatory and strictly sign-regular matrices, Lin. Alg. and Appl. 246, 17–21.CrossRefGoogle Scholar
Fallat, S. M., Gekhtman, M. I. and Johnson, C. R. [2000], Spectral structures of irreducible totally nonnegative matrices, SIAM J. Matrix Anal. Appl. 22, 627–645.CrossRefGoogle Scholar
Fallat, S. M., Gekhtman, M. I. and Johnson, C. R. [2003], Multiplicative principal-minor inequalities for totally nonnegative matrices, Adv. Applied Math. 30, 442–470.CrossRefGoogle Scholar
Fan, K. [1966], Some matrix inequalities, Abh. Math. Sem. Univ. Hamburg 29, 185–196.CrossRefGoogle Scholar
Fan, K. [1967], Subadditive functions on a distributive lattice and an extension of Szasz's inequality, J. Math. Anal. Appl. 18, 262–268.CrossRefGoogle Scholar
Fan, K. [1968], An inequality for subadditive functions on a distributive lattice with application to determinantal inequalities, Lin. Alg. and Appl. 1, 33–38.CrossRefGoogle Scholar
Fekete, M. and Pólya, G. [1912], Über ein Problem von Laguerre, Rend. C. M. Palermo 34, 89–120.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A. [1999], Double Bruhat cells and total positivity, J. Amer. Math. Soc. 12, 335–380.CrossRefGoogle Scholar
Fomin, S. and Zelevinsky, A. [2000], Total positivity: tests and parametrizations, Math. Intell. 22, 23–33.CrossRefGoogle Scholar
Friedland, S. [1985], Weak interlacing properties of totally positive matrices, Lin. Alg. and Appl. 71, 95–100.CrossRefGoogle Scholar
Gantmacher, F. [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
Gantmacher, F. R. [1953], The Theory of Matrices, Gostekhizdat, Moscow- Leningrad; English transl. as Matrix Theory, Chelsea, New York, 2 vols., 1959.Google Scholar
Gantmacher, F. R. [1965], Obituary, in Uspekhi Mat. Nauk 20, 149–158; English transl. as Russian Math. Surveys, 20(1965), 143–151.
Gantmacher, F. R. and Krein, M. G. [1935], Sur les matrices oscillatoires, C. R. Acad. Sci.(Paris) 201, 577–579.Google Scholar
Gantmakher, F. R. and Krein, M. G. [1937], Sur les matrices compl`etement non négatives et oscillatoires, Compositio Math. 4, 445–476.Google Scholar
Gantmacher, F. R. and Krein, M. G. [1941], Oscillation Matrices and Small Oscillations of Mechanical Systems (Russian), Gostekhizdat, Moscow-Leningrad.
Gantmacher, F. R. and Krein, M. G. [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.
Garloff, J. [1982], Criteria for sign regularity of sets of matrices, Lin. Alg. and Appl. 44, 153–160.CrossRefGoogle Scholar
Garloff, J. [2003], Intervals of almost totally positive matrices, Lin. Alg. and Appl. 363, 103–108.CrossRefGoogle Scholar
Garloff, J. and Wagner, D. [1996a], Hadamard products of stable polynomials are stable, J. Math. Anal. Appl. 202, 797–809.CrossRefGoogle Scholar
Garloff, J. and Wagner, D. [1996b], Preservation of total nonnegativity under the Hadamard products and related topics, in Total Positivity and its Applications eds., Micchelli, C. A. and Gasca, M., Kluwer Acad. Publ., Dordrecht, 97–102.CrossRefGoogle Scholar
Gasca, M. and Micchelli, C. A. [1996], Total Positivity and its Applications, eds., Kluwer Acad. Publ., Dordrecht.CrossRefGoogle Scholar
Gasca, M., Micchelli, C. A. and Peña, J. M. [1992], Almost strictly totally positive matrices, Numerical Algorithms 2, 225–236.CrossRefGoogle Scholar
Gasca, M. and Peña, J. M. [1992], Total positivity and Neville elimination, Lin. Alg. and Appl. 165, 25–44.CrossRefGoogle Scholar
Gasca, M. and Peña, J. M. [1993], Total positivity, QR factorization, and Neville elimination, SIAM J. Matrix Anal. Appl. 14, 1132–1140.CrossRefGoogle Scholar
Gasca, M. and Peña, J. M. [1995], On the characterization of almost strictly totally positive matrices, Adv. Comp. Math. 3, 239–250.CrossRefGoogle Scholar
Gasca, M. and Peña, J. M. [1996], On factorizations of totally positive matrices, in Total Positivity and its Applications eds., Micchelli, C. A. and Gasca, M., Kluwer Acad. Publ., Dordrecht, 109–130.CrossRefGoogle Scholar
Gasca, M. and Peña, J. M. [2006], Characterizations and decompositions of almost strictly totally positive matrices, SIAM J. Matrix Anal. 28, 1–8.CrossRefGoogle Scholar
Gladwell, G. M. L. [1998], Total positivity and the QR algorithm, Lin. Alg. and Appl. 271, 257–272.CrossRefGoogle Scholar
Gladwell, G. M. L. [2004], Inner totally positive matrices, Lin. Alg. and Appl. 393, 179–195.CrossRefGoogle Scholar
Gohberg, I. [1989], Mathematical Tales, in The Gohberg Anniversary Collection, Eds. Dym, H., Goldberg, S., Kaashoek, M. A., Lancaster, P., pp. 17–56, Operator Theory: Advances and Applications, Vol. 40, Birkhäuser Verlag, Basel.CrossRefGoogle Scholar
Gohberg, I. [1990], Mark Grigorievich Krein 1907–1989, Notices Amer. Math. Soc. 37, 284–285.Google Scholar
Goodman, T. N. T. [1995], Total positivity and the shape of curves, in Total Positivity and its Applications eds., Micchelli, C. A. and Gasca, M., Kluwer Acad. Publ., Dordrecht, 157–186.Google Scholar
Goodman, T. N. T. and Sun, Q. [2004], Total positivity and refinable functions with general dilation, Appl. Comput. Harmon. Anal. 16, 69–89.CrossRefGoogle Scholar
Gross, K. I. and Richards, D. St. P. [1995], Total positivity, finite reflection groups, and a formula of Harish-Chandra, J. Approx. Theory 82, 60–87.CrossRefGoogle Scholar
Holtz, O. [2003], Hermite-Biehler, Routh-Hurwitz and total positivity, Lin. Alg. and Appl. 372, 105–110.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R. [1991], Topics in Matrix Analysis, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Karlin, S. [1964], The existence of eigenvalues for integral operators, Trans. Amer. Math. Soc. 113, 1–17.CrossRefGoogle Scholar
Karlin, S. [1965], Oscillation properties of eigenvectors of strictly totally positive matrices, J. D'Analyse Math. 14, 247–266.CrossRefGoogle Scholar
Karlin, S. [1968], Total Positivity. Volume 1, Stanford University Press, Stanford, CA.Google Scholar
Karlin, S. [1972], Some extremal problems for eigenvalues of certain matrix and integral operators, Adv. in Math. 9, 93–136.CrossRefGoogle Scholar
Karlin, S. [2002], Interdisciplinary meandering in science. 50th anniversary issue of Operations Research. Oper. Res. 50, 114–121.CrossRefGoogle Scholar
Karlin, S. and Pinkus, A. [1974], Oscillation properties of generalized characteristic polynomials for totally positive and positive definite matrices, Lin. Alg. and Appl. 8, 281–312.CrossRefGoogle Scholar
Karlin, S. and Studden, W. J. [1966], Tchebycheff Systems: with Applications in Analysis and Statistics, Interscience Publishers, John Wiley, New York.Google Scholar
Katkova, O. M. and Vishnyakova, A. M. [2006], On sufficient conditions for the total positivity and for the multiple positivity of matrices, Lin. Alg. and Appl. 416, 1083–1097.CrossRefGoogle Scholar
Kellogg, O. D. [1918], Orthogonal function sets arising from integral equations, Amer. J. Math. 40, 145–154.CrossRefGoogle Scholar
Kellogg, O. D. [1929], Foundations of Potential Theory, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Kemperman, J. H. B. [1982], A Hurwitz matrix is totally positive, SIAM J. Math. Anal. 13, 331–341.CrossRefGoogle Scholar
Koev, P. [2005], Accurate eigenvalues and SVDs of totally nonnegative matrices, SIAM J. Matrix. Anal. Appl. 27, 1–23.CrossRefGoogle Scholar
Koev, P. [2007], Accurate computations with totally nonnegative matrices, SIAM J. Matrix. Anal. Appl. 29, 731–751.CrossRefGoogle Scholar
Koteljanskii, D. M. [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
Koteljanskii, D. M. [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
Krein, M. G. and Nudel'man, A. A. [1977], The Markov Moment Problem and Extremal Problems, Transl. Math. Monographs, Vol. 50, Amer. Math. Society, Providence.CrossRef
Kurtz, D. C. [1992], A sufficient condition for all the roots of a polynomial to be real, Amer. Math. Monthly 99, 259–263.CrossRefGoogle Scholar
Loewner, K. [1955], On totally positive matrices, Math. Z. 63, 338–340.CrossRefGoogle Scholar
Lusztig, G. [1994], Total positivity in reductive groups, in Lie Theory and Geometry, Progress in Mathematics, 123, Birkhäuser, Boston, 531–568.Google Scholar
Maló, E. [1895], Note sur les équations algébriques dont toutes les racines sont réelles, Journal de Mathématiques Spéciales 4, 7–10.Google Scholar
Marden, M. [1966], Geometry of Polynomials, Math. Surveys, Vol. 3 (2nd edition), Amer. Math. Society, Providence.
Markham, T. L. [1970], A semi-group of totally nonnegative matrices, Lin. Alg. and Appl. 3, 157–164.CrossRefGoogle Scholar
Marshall, A. W. and Olkin, I. [1979], Inequalities: Theory of Majorization and its Applications, Academic Press, New York.Google Scholar
Metelmann, K. [1973], Ein Kriterium für den Nachweis der Totalnichtnegativität von Bandmatrizen, Lin. Alg. and Appl. 7, 163–171.CrossRef
Micchelli, C. A. and Pinkus, A. [1991], Descartes systems from corner cutting, Constr. Approx. 7, 161–194.CrossRefGoogle Scholar
Motzkin, Th. [1936], Beiträge zur Theorie der linearen Ungleichungen, Doctoral Dissertation, Basel, 1933. Azriel Press, Jerusalem.Google Scholar
Mühlbach, G. and Gasca, M. [1985], A generalization of Sylvester's identity on determinants and some applications, Lin. Alg. and Appl. 66, 221–234.CrossRefGoogle Scholar
Pinkus, A. [1985a], Some extremal problems for strictly totally positive matrices, Lin. Alg. and Appl. 64, 141–156.CrossRefGoogle Scholar
Pinkus, A. [1985b], n-widths of Sobolev spaces in Lp, Const. Approx. 1, 15–62.CrossRef
Pinkus, A. [1985c], n-Widths in Approximation Theory, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Pinkus, A. [1996], Spectral properties of totally positive kernels and matrices, in Total Positivity and its Applications eds., Micchelli, C. A. and Gasca, M., Kluwer Acad. Publ., Dordrecht, 477–511.CrossRefGoogle Scholar
Pinkus, A. [1998], An interlacing property of eigenvalues of strictly totally positive matrices, Lin. Alg. and Appl. 279, 201–206.CrossRefGoogle Scholar
Pinkus, A. [2008], Zero minors of totally positive matrices, Electronic J. Linear Algebra 17, 532–542.CrossRefGoogle Scholar
Pitman, J. [1997], Probabilistic bounds on the coefficients of polynomials with only real zeros, J. Comb. Theory, Ser. A 77, 279–303.CrossRefGoogle Scholar
Pólya, G. and Szegő, G. [1976], Problems and Theorems in Analysis II, Springer-Verlag, New York.CrossRefGoogle Scholar
Rahman, Q. I. and Schmeisser, G. [2002], Analytic Theory of Polynomials, Oxford University Press, Oxford.Google Scholar
Schoenberg, I. J. [1930], Über variationsvermindernde lineare Transformationen, Math. Z. 32, 321–328.CrossRefGoogle Scholar
Schoenberg, I. J. and Whitney, A. [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
Schumaker, L. L. [1981], Spline Functions: Basic Theory, John Wiley & Sons, New York.Google Scholar
Shapiro, B. Z. and Shapiro, M. Z. [1995], On the boundary of totally positive upper triangular matrices, Lin. Alg. and Appl. 231, 105–109.CrossRefGoogle Scholar
Shohat, J. A. and Tamarkin, J. D. [1943], The Problem of Moments, Math. Surveys, Vol. 1, Amer. Math. Society, Providence.CrossRef
Skandera, M. [2004], Inequalities in products of minors of totally nonnegative matrices, J. Alg. Comb. 20, 195–211.CrossRefGoogle Scholar
Smith, P. W. [1983], Truncation and factorization of bi-infinite matrices, in Approximation Theory, IV (College Station, Tex., 1983), 257–289, Academic Press, New York.Google Scholar
Stieltjes, T. J. [1894–95], Recherches sur les fractions continues, Annales Fac. Sciences Toulouse 8, 1–122; 9, 1–47.CrossRefGoogle Scholar
Wagner, D. [1992], Total positivity of Hadamard products, J. Math. Anal. Appl. 163, 459–483.CrossRefGoogle Scholar
Wang, Y. and Yeh, Y.-N. [2005], Polynomials with real zeros and Pólya frequency sequences, J. Comb. Theory, Ser. A 109, 63–74.CrossRefGoogle Scholar
Whitney, A. [1952], A reduction theorem for totally positive matrices, J. d'Anal. Math. 2, 88–92.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Totally Positive Matrices
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511691713.009
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Totally Positive Matrices
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511691713.009
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • References
  • Allan Pinkus, Technion - Israel Institute of Technology, Haifa
  • Book: Totally Positive Matrices
  • Online publication: 05 May 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511691713.009
Available formats
×