Stability and inertia

Abstract The purpose of this paper is to present a brief overview of matrix stability and inertia theory. A few applications of inertia and stability theorems, and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented. Inter-relationships between different theorems are explicitly stated, whenever appropriate. The paper concludes with some problems for future research in this area.

[1]  P. Parks,et al.  Liapunov and the Schur-Cohn stability criterion , 1964 .

[2]  B. Datta.  Application of Hankel matrices of Markov Parameters to the solutions of the Routh-Hurwitz and the Schur-Cohn problems , 1979 .

[3]  H. Wimmer,et al.  An inertia theorem for tridiagonal matrices and a criterion of wall on continued fractions , 1974 .

[4]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[5]  B. Cain,et al.  The inertial aspects of Stein’s condition -*≫ , 1974 .

[6]  Biswa Nath Datta,et al.  On the Routh-Hurwitz-Fujiwara and the Schur-Cohn-Fujiwara theorems for the root-separation problem , 1978 .

[7]  M. Kreĭn,et al.  Stability of Solutions of Differential Equations in Banach Spaces , 1974 .

[8]  M. Marden.  Geometry of Polynomials , 1970 .

[9]  Harald K. Wimmer,et al.  Remarks on inertia theorems for matrices , 1975 .

[10]  D. Carlson.  On the Controllability of Matrix Pairs $(A, K) with $K$ Positive Semidefinite, II , 1994 .

[11]  Chi-Tsong Chen,et al.  Inertia theorem for general matrix equations , 1975 .

[12]  M. Tismenetsky,et al.  Generalized Bezoutian and matrix equations , 1988 .

[13]  K. Arrow,et al.  A NOTE ON DYNAMIC STABILITY , 1958 .

[14]  Some extensions and modifications of classical stability tests for polynomials , 1983 .

[15]  Hans Schneider,et al.  Inertia theorems for matrices: The semidefinite case , 1963 .

[16]  M. Tismenetsky,et al.  Bezoutian and Schur-Cohn problem for operator polynomials , 1984 .

[17]  Mark J. Balas,et al.  Trends in large space structure control theory: Fondest hopes, wildest dreams , 1982 .

[18]  V. Pták,et al.  Zero location by Hermitian forms: The singular case , 1982 .

[19]  David Carlson,et al.  Controllability, inertia, and stability for tridiagonal matrices , 1984 .

[20]  O. Taussky.  Matrices C with Cn → 0 , 1964 .

[21]  R. Kálmán.  Algebraic characterization of polynomials whose zeros lie in certain algebraic domains. , 1969, Proceedings of the National Academy of Sciences of the United States of America.

[22]  W. Hahn.  Eine Bemerkung zur zweiten Methode von LJAPUNOV , 1955 .

[23]  B. Anderson.  Application of the Second Method of Lyapunov to the Proof of the Markov Stability Criterion , 1967 .

[24]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[25]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[26]  B. Datta.  An elementary proof of the stability criterion of Liénard and Chipart , 1978 .

[27]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[28]  L. Rodman,et al.  Inertia Theorems for Hilbert Space Operators Based on Lyapunov and Stein Equations , 1999 .

[29]  P. Stein.  Some general theorems on iterants , 1952 .

[30]  S. Bialas.  On the Lyapunov matrix equation , 1980 .

[31]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[32]  Harald K. Wimmer,et al.  Inertia theorems for matrices, controllability, and linear vibrations , 1974 .

[33]  J. Bunch,et al.  Decomposition of a symmetric matrix , 1976 .

[34]  Alston S. Householder,et al.  Bezoutiants, Elimination and Localization , 1970 .

[35]  K. Glover.  All optimal Hankel-norm approximations of linear multivariable systems and their L, ∞ -error bounds† , 1984 .

[36]  B. Datta.  Stability and D-stability , 1978 .

[37]  Harald K. Wimmer,et al.  On the algebraic Riccati equation , 1976, Bulletin of the Australian Mathematical Society.

[38]  P. Parks.  A new proof of the Routh-Hurwitz stability criterion using the second method of Liapunov , 1962, Mathematical Proceedings of the Cambridge Philosophical Society.

[39]  George Cybenko,et al.  Fast toeplitz orthogonalization using inner decompositions , 1987 .

[40]  B. Datta,et al.  Feedback stabilization of a second-order system: A nonmodal approach , 1993 .

[41]  D. Carlson,et al.  Controllability and inertia theory for functions of a matrix , 1977 .

[42]  B. Cain.  An inertia theory for operators on a Hilbert space , 1973 .

[43]  R. D. Hill,et al.  Inertia theory for simultaneously triangulable complex matrices , 1969 .

[44]  B. Datta,et al.  On finding eigenvalue distribution of a matrix in several regions of the complex plane , 1984, The 23rd IEEE Conference on Decision and Control.

[45]  E. Jury,et al.  Remarks on the root-clustering of a polynomial in a certain region in the complex plane , 1974 .

[46]  Brian D. O. Anderson,et al.  Schwarz matrix properties for continuous and discrete time systems , 1976 .

[47]  Dragoslav D. Šiljak,et al.  Alexandr Michailovich Liapunov (1857 – 1918) , 1976 .

[48]  A. Berman,et al.  Positive diagonal solutions to the Lyapunov equations , 1978 .

[49]  S. Barnett,et al.  A note on matrix equations and root location , 1975 .

[50]  Biswa Nath Datta,et al.  The Lyapunov matrix equation SA+A∗S=S∗B∗BS , 1979 .

[51]  D. Carlson,et al.  On Ranges of Lyapunov Transformations. , 1974 .

[52]  W. Schmitendorf,et al.  A Simple Test for Asymptotic Stability in Partially Dissipative Symmetric Systems , 1973 .

[53]  Abraham Berman,et al.  Matrix Diagonal Stability and Its Implications , 1983 .

[54]  Paul Van Dooren,et al.  Structured linear algebra problems in digital signal processing , 1991 .

[55]  Biswa Nath Datta,et al.  On Bezoutians, Van der Monde matrices, and the Lienard-Chipart stability criterion , 1989 .

[56]  Amit Bhaya,et al.  On the design of large flexible space structures(LFSS) , 1985, 1985 24th IEEE Conference on Decision and Control.

[57]  Charles R. Johnson.  Sufficient conditions for D-stability , 1974 .

[58]  Charles R. Johnson,et al.  A Semi-Definite Lyapunov Theorem and the Characterization of Tridiagonal D-Stable Matrices , 1982 .

[59]  E. Ecer,et al.  Numerical Linear Algebra and Applications , 1995, IEEE Computational Science and Engineering.

[60]  B. O. Anderson,et al.  Generalized Bezoutian and Sylvester matrices in multivariable linear control , 1976, 1976 IEEE Conference on Decision and Control including the 15th Symposium on Adaptive Processes.

[61]  H. Schneider.  Positive operators and an inertia theorem , 1965 .

[62]  P. Lancaster,et al.  Inertia characteristics of self-adjoint matrix polynomials , 1983 .

[63]  Biswa Nath Datta,et al.  Linear and numerical linear algebra in control theory: some research problems , 1994 .

[64]  Raphael Loewy,et al.  An inertia theorem for Lyapunov's equation and the dimension of a controllability space , 1997 .

[65]  J. Bunch.  Analysis of the Diagonal Pivoting Method , 1971 .

[66]  M.L.J. Hautus,et al.  Controllability and observability conditions of linear autonomous systems , 1969 .

[67]  Y Saad,et al.  Numerical methods for large eigenvalue problems : theory and algorithms , 1992 .

[68]  Similarity and the numerical range , 1969 .

[69]  Biswa Nath Datta,et al.  The matrix equation XA=ATX and an associated algorithm for solving the inertia and stability problems☆ , 1987 .

[70]  Bryan E. Cain.  The Inertial Aspects of Stein's Condition H - C ∗ HC ≫0 , 1974 .

[71]  R. D. Hill,et al.  Eigenvalue location using certain matrix functions and geometric curves , 1977 .

[72]  A. Liapounoff,et al.  Problème général de la stabilité du mouvement , 1907 .

[73]  Ole Østerby,et al.  Direct Methods for Space Matrices , 1983, Lecture Notes in Computer Science.

[74]  Chi-Tsong Chen,et al.  A Generalization of the Inertia Theorem , 1973 .

[75]  S. Liberty,et al.  Linear Systems , 2010, Scientific Parallel Computing.

[76]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[77]  H. Wielandt,et al.  On the eigenvalues of A + B and AB , 1973 .

[78]  S. Gutman,et al.  A general theory for matrix root-clustering in subregions of the complex plane , 1981 .

[79]  D. Carlson,et al.  On the effective computation of the inertia of a non-hermitian matrix , 1979 .

[80]  S. Lehnigk.  Liapunov’s Direct Method and the Number of Zeros with Positive Real Parts of a Polynomial with Constant Complex Coefficients , 1967 .

[81]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[82]  G. F. Bryant.  On a generalization of a theorem of Lyapunov , 1975 .

[83]  T. Kailath,et al.  Generalized Displacement Structure for Block-Toeplitz,Toeplitz-Block, and Toeplitz-Derived Matrices , 1994 .

[84]  Patrizio Colaneri,et al.  Lyapunov and Riccati equations: Periodic inertia theorems , 1986 .

[85]  H. Wimmer,et al.  On the Ostrowski-Schneider Inertia Theorem , 1973 .

[86]  S. Barnett,et al.  A Note on the Bezoutian Matrix , 1972 .

[87]  S. K. Mitter,et al.  Algebraic methods for the solution of some linear matrix equations , 1982 .

[88]  J. L. Howland,et al.  Matrix equations and the separation of matrix eigenvalues , 1971 .

[89]  B. Datta,et al.  Matrix equation, matrix polynomial and the number of zeros of a polynomial inside the unit circle , 1980 .

[90]  S. K. Mitter,et al.  Algebraic Methods for the Study of Some Linear Matrix Equations , 2001 .

[91]  M. Tismenetsky,et al.  The Bezoutian and the eigenvalue-separation problem for matrix polynomials , 1982 .

[92]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[93]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[94]  A. M. Li︠a︡punov.  Problème général de la stabilité du mouvement , 1949 .

[95]  Siegfried H Lehnigk,et al.  Stability theorems for linear motions : with an introduction to Liapunov's direct method , 1966 .

[96]  Carmen Chicone,et al.  A generalization of the inertia theorem for quadratic matrix polynomials , 1998 .

[97]  Mark A. Shayman,et al.  Inertia theorems for the periodic Liapunov equation and periodic Riccati equation , 1984 .

[98]  Abraham Berman,et al.  Characterization of acyclic d-stable matrices☆ , 1984 .

[99]  S. Barnett,et al.  Matrices, polynomials, and linear time-variant systems , 1973 .

[100]  Beresford N. Parlett,et al.  Use of indefinite pencils for computing damped natural modes , 1990 .

[101]  Antony Jameson,et al.  Solution of the Equation $AX + XB = C$ by Inversion of an $M \times M$ or $N \times N$ Matrix , 1968 .

[102]  M. Fujiwara,et al.  Über die algebraischen Gleichungen, deren Wurzeln in einem Kreise oder in einer Halbebene liegen , 1926 .

[103]  J. Snyders,et al.  On Nonnegative Solutions of the Equation $AD + DA' = - C$ , 1970 .

[104]  Adam W. Bojanczyk,et al.  On the stability of the Bareiss and related Toeplitz factorization algorithms , 2010, SIAM J. Matrix Anal. Appl..

[105]  M. Naimark,et al.  The method of symmetric and Hermitian forms in the theory of the separation of the roots of algebraic equations , 1981 .

[106]  Lyapunov Revisited: Variations on a Matrix Theme , 1997 .