A Survey of Classes of Matrices Possessing the Interval Property and Related Properties

This paper considers intervals of real matrices with respect to partial orders and the problem to infer from some exposed matrices lying on the boundary of such an interval that all real matrices taken from the interval possess a certain property. In many cases such a property requires that the chosen matrices have an identically signed inverse. We also briefly survey related problems, e.g., the invariance of matrix properties under entry-wise perturbations.

[1]  J. Urgen Garloo,et al.  Intervals of Almost Totally Positive Matrices , 1984 .

[2]  E. Kaszkurewicz,et al.  Matrix diagonal stability in systems and computation , 1999 .

[3]  S. P. Eveson,et al.  Totally positive matrices (Cambridge Tracts in Mathematics 181) , 2011 .

[4]  Jiri Rohn,et al.  Inverse interval matrix , 1993 .

[5]  J. Kuttler,et al.  A fourth-order finite-difference approximation for the fixed membrane eigenproblem , 1971 .

[6]  J. Garloff,et al.  Total nonnegativity of matrices related to polynomial roots and poles of rational functions , 2016 .

[7]  Jürgen Garloff,et al.  Criteria for sign regularity of sets of matrices , 1982 .

[8]  D. Hertz.  The extreme eigenvalues and stability of real symmetric interval matrices , 1992 .

[9]  Jürgen Garloff,et al.  Vertex Implications for Totally Nonnegative Matrices , 1996 .

[10]  Stéphane Launois,et al.  Efficient Recognition of Totally Nonnegative Matrix Cells , 2014, Found. Comput. Math..

[11]  J. Garloff,et al.  Intervals of totally nonnegative matrices , 2013 .

[12]  Invariance of Total Positivity of a Matrix under Entry-wise Perturbation and Completion Problems , 2016 .

[13]  Vladik Kreinovich,et al.  Optimal Finite Characterization of Linear Problems with Inexact Data , 2005, Reliab. Comput..

[14]  Wilhelm Barth,et al.  Optimale Lösung von Intervallgleichungssystemen , 2005, Computing.

[15]  Allan Pinkus.  Totally Positive Matrices , 2009 .

[16]  J. Rohn.  Forty necessary and sufficient conditions for regularity of interval matrices: A survey , 2009 .

[17]  Rama K. Yedavalli,et al.  Sufficiency of vertex matrix check for robust stability of interval matrices via the concept of Qualitative Robustness , 2013, 2013 European Control Conference (ECC).

[18]  F. R. Gantmakher,et al.  Oszillationsmatrizen, Oszillationskerne und kleine Schwingungen mechanischer Systeme , 1960 .

[19]  J. Garloff,et al.  IMPROVED TESTS AND CHARACTERIZATIONS OF TOTALLY NONNEGATIVE MATRICES , 2014 .

[20]  Takehiro Mori,et al.  Convergence property of interval matrices and interval polynomials , 1987 .

[21]  Ronald L. Smith,et al.  Intervals of Inverse M-Matrices , 2002, Reliab. Comput..

[22]  Jürgen Garloff.  Pivot tightening for direct methods for solving symmetric positive definite systems of linear interval equations , 2011, Computing.

[23]  A. Tesi,et al.  New conditions for global stability of neural networks with application to linear and quadratic programming problems , 1995 .

[24]  K. Metelmann.  Ein Kriterium für den Nachweis der Totalnichtnegativität von Bandmatrizen , 1973 .

[25]  J. Garloff,et al.  Intervals of P-matrices and related matrices , 1984 .

[26]  Michael K. Ng,et al.  Maximum inverse positive matrices , 2007, Appl. Math. Lett..

[27]  Tingzhu Huang,et al.  The inverse positivity of perturbed tridiagonal M-matrices☆ , 2011 .

[28]  J. Rohn.  Systems of linear interval equations , 1989 .

[29]  Jürgen Garloff,et al.  Boundary implications for stability properties: present status , 1988 .

[30]  J. Garloff,et al.  Intervals of special sign regular matrices , 2016 .

[31]  Charles R. Johnson.  Inverse M-matrices☆ , 1982 .

[32]  Kurt Metelmann.  Inverspositive Bandmatrizen und totalnichtnegative Green'sche Matrizen , 1972 .

[33]  Jianzhou Liu,et al.  Nonsingular almost strictly sign regular matrices , 2012 .

[34]  Mohammad Adm.  Perturbation and Intervals of Totally Nonnegative Matrices and Related Properties of Sign Regular Matrices , 2016 .

[35]  Shaun M. Fallat,et al.  Totally Nonnegative Matrices , 2011 .

[37]  J. Rohn.  Positive Definiteness and Stability of Interval Matrices , 1994, SIAM J. Matrix Anal. Appl..

[38]  Jie Huang,et al.  Monotonicity of Perturbed Tridiagonal M-Matrices , 2012, SIAM J. Matrix Anal. Appl..

[39]  Charles R. Johnson,et al.  Inverse M-matrices, II , 2011 .

[40]  K. Goodearl,et al.  Totally nonnegative cells and matrix Poisson varieties , 2009, 0905.3631.