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In this paper, we consider how to accurately solve the product eigenvalue problem for the class of sign regular (SR) matrices with signature (1 , ⋯ , 1 , − 1) and the class of totally nonnegative (TN) matrices, which tend to be extremely ill-conditioned. We present algorithms with O (n 3) complexity to accurately compute the parameter matrices of products of TN matrices and SR matrices with signature (1 , ... , 1 , − 1). Based on the accurate parameter matrices, all eigenvalues of the product matrix are computed to high relative accuracy. Numerical experiments are provided to confirm the claimed high relative accuracy. [ABSTRACT FROM AUTHOR]

A real linear combination of products of minors which is nonnegative over all totally nonnegative (TN) matrices is called a determinantal inequality for these matrices. It is referred to as multiplicative when it compares two collections of products of minors and additive otherwise. Set theoretic operations preserving the class of TN matrices naturally translate into operations preserving determinantal inequalities in this class. We introduce index-row (and index-column) operations that act directly on all determinantal inequalities for TN matrices, and yield further inequalities for these matrices. These operations assist in revealing novel additive inequalities for TN matrices embedded in the classical identities due to Laplace [ Mem. Acad. Sciences Paris 1772] and Karlin (1968). In particular, for any square TN matrix A , these derived inequalities generalize – to every i th row of A and j th column of adj A – the classical Gantmacher–Krein fluctuating inequalities (1941) for i = j = 1. Furthermore, our index-row/column operations reveal additional undiscovered fluctuating inequalities for TN matrices. The introduced index-row/column operations naturally birth an algorithm that can detect certain determinantal expressions that do not form an inequality for TN matrices. However, the algorithm completely characterizes the multiplicative inequalities comparing products of pairs of minors. Moreover, the underlying index-row/column operations add that these inequalities are offshoots of certain "complementary/higher" ones. These novel results seem very natural, and in addition thoroughly describe and enrich the classification of these multiplicative inequalities due to Fallat–Gekhtman–Johnson [ Adv. Appl. Math. 2003] and later Skandera [ J. Algebraic Comb. 2004] (and revisited by Rhoades–Skandera [ Ann. Comb. 2005, J. Algebra 2006]). [ABSTRACT FROM AUTHOR]

Recently, a spectral Favard theorem was presented for bounded banded lower Hessenberg matrices that possess a positive bidiagonal factorization. The paper establishes conditions, expressed in terms of continued fractions, under which an oscillatory tetradiagonal Hessenberg matrix can have such a positive bidiagonal factorization. Oscillatory tetradiagonal Toeplitz matrices are examined as a case study of matrices that admit a positive bidiagonal factorization. Furthermore, the paper proves that oscillatory banded Hessenberg matrices are organized in rays, where the origin of the ray does not have a positive bidiagonal factorization, but all the interior points of the ray do have such a positive bidiagonal factorization. [ABSTRACT FROM AUTHOR]

Recently a spectral Favard theorem for bounded banded lower Hessenberg matrices that admit a positive bidiagonal factorization was presented. These type of matrices are oscillatory. In this paper the Lima–Loureiro hypergeometric multiple orthogonal polynomials and the Jacobi–Piñeiro multiple orthogonal polynomials are discussed at the light of this bidiagonal factorization for tetradiagonal matrices. The Darboux transformations of tetradiagonal Hessenberg matrices is studied and Christoffel formulas for the elements of the bidiagonal factorization are given, i.e., the bidiagonal factorization is given in terms of the recursion polynomials evaluated at the origin. [ABSTRACT FROM AUTHOR]

The matrix S = [1+xiyj]i, j=1n, 0 < x1 < · · · < xn, 0 < y1 < · · · < yn, has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of S in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition. We give a bidiagonal decomposition of S◦m = [(1+xiyj)m] for positive integers 1 ≤ m ≤ n-1. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices. [ABSTRACT FROM AUTHOR]

Oscillatory matrices were introduced in the seminal work of Gantmacher and Krein. An n × n matrix A is called oscillatory if all its minors are nonnegative and there exists a positive integer k such that all minors of A k are positive. The smallest k for which this holds is called the exponent of the oscillatory matrix A. Gantmacher and Krein showed that the exponent is always smaller than or equal to n − 1. An important and nontrivial problem is to determine the exact value of the exponent. Here we use the successive elementary bidiagonal factorization of oscillatory matrices, and its graph-theoretic representation, to derive an explicit expression for the exponent of several classes of oscillatory matrices, and a nontrivial upper-bound on the exponent for several other classes. [ABSTRACT FROM AUTHOR]

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given. [ABSTRACT FROM AUTHOR]

The space of m× p totally nonnegative real matrices has a stratification into totally nonnegative cells. The largest such cell is the space of totally positive matrices. There is a well-known criterion due to Gasca and Peña for testing a real matrix for total positivity. This criterion involves testing mp minors. In contrast, there is no known small set of minors for testing for total nonnegativity. In this paper, we show that for each of the totally nonnegative cells there is a test for membership which only involves mp minors, thus extending the Gasca and Peña result to all totally nonnegative cells. [ABSTRACT FROM AUTHOR]

Abstract: Let be a polynomial with positive coefficients and . The pth Hadamard power of is the polynomial . It is conjectured that if has only real zeros, then so does for . We verify the conjecture when and give a counterexample when . We also show that there exists a positive number such that if has only real zeros, then so does for . [Copyright &y& Elsevier]

Abstract: For the first time, perturbation bounds including componentwise perturbation bounds for the block LU factorization have been provided by Dopico and Molera (2005) [5]. In this paper, componentwise error analysis is presented for computing the block LU factorization of nonsingular totally nonnegative matrices. We present a componentwise bound on the equivalent perturbation for the computed block LU factorization. Consequently, combining with the componentwise perturbation results we derive componentwise forward error bounds for the computed block factors. [Copyright &y& Elsevier]

Abstract: A uniqueness theorem for an LU decomposition of a totally nonnegative matrix is obtained. [Copyright &y& Elsevier]

Abstract: We describe explicitly the admissible families of minors for the totally nonnegative cells of real matrices, that is, the families of minors that produce nonempty cells in the cell decompositions of spaces of totally nonnegative matrices introduced by A. Postnikov. In order to do this, we relate the totally nonnegative cells to torus orbits of symplectic leaves of the Poisson varieties of complex matrices. In particular, we describe the minors that vanish on a torus orbit of symplectic leaves, we prove that such families of minors are exactly the admissible families, and we show that the nonempty totally nonnegative cells are the intersections of the torus orbits of symplectic leaves with the spaces of totally nonnegative matrices. [ABSTRACT FROM AUTHOR]

Abstract: We show that modified Richardson method converges for any nonsingular totally nonnegative stochastic matrix for any choice of the parameter between 0 and 2. We present a variant of the modified Richardson method that is convergent for any nonsingular totally nonnegative matrix. We obtain the optimal parameter value for this method and give a procedure for estimating it. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

Abstract: The issue regarding Schur complements of sign regular matrices is rather subtle. It is known that the class of totally nonnegative matrices is not closed under arbitrary Schur complementation. In this paper, we demonstrate how Schur complements of sign regular matrices of order are sign regular of a certain order. In particular, some results for totally nonnegative and totally nonpositive matrices are provided as our corollaries. [Copyright &y& Elsevier]

Abstract: We investigate -matrices which are totally nonnegative and therefore which have all of their eigenvalues equal to nonnegative real numbers. Such matrices are characterized by four forbidden submatrices (of orders 2 and 3). We show that the maximum number of s in an irreducible, totally nonnegative -matrix of order is and characterize those matrices with this number of s. We also show that the minimum Perron value of an irreducible, totally nonnegative -matrix of order equals and characterize those matrices with this Perron value. [Copyright &y& Elsevier]

Abstract: A matrix A is called an oscillatory matrix if it is totally nonnegative and there exists a positive integer k such that A k is totally positive. In this paper we describe the class of n × n oscillatory matrices which have maximal exponent equal to n −1. [Copyright &y& Elsevier]

Abstract: We present a new algorithm for computing all eigenvalues of certain sign regular matrices to high relative accuracy in floating point arithmetic. The accuracy and cost are unaffected by the conventional eigenvalue condition numbers. A matrix is called sign regular when the signs of its nonzero minors depend only of the order of the minors. The sign regular matrices we consider are the ones which are nonsingular and whose kth order nonzero minors are of sign for all k. This class of matrices can also be characterized as “nonsingular totally nonnegative matrices with columns in reverse order”. We exploit a characterization of these particular sign regular matrices as products of nonnegative bidiagonals and the reverse identity. We arrange the computations in such a way that no subtractive cancellation is encountered, thus guaranteeing high relative forward accuracy. [Copyright &y& Elsevier]

Abstract: Considered are continuous, positive Hadamard powers of entry-wise positive (nonnegative) matrices. Those that are eventually (in the sense of all Hadamard powers beyond some point) totally positive, totally nonnegative, doubly nonnegative and doubly positive are characterized. For example, for matrices with at least four rows and four columns, Hadamard powers greater than one of totally positive matrices need not be totally positive, but they are eventually totally positive. [Copyright &y& Elsevier]

In this note we revisit a classical criterion obtained by Gantmacher and Krein for determining when a totally nonnegative matrix is actually oscillatory. A new proof of this criterion is presented by incorporating bidiagonal factorizations of invertible totally nonnegative matrices and utilizing certain associated weighted planar diagrams. [Copyright &y& Elsevier]

Simple proofs of the Hermite–Biehler and Routh–Hurwitz theorems are presented. The total nonnegativity of the Hurwitz matrix of a stable real polynomial follows as an immediate corollary. [Copyright &y& Elsevier]

An m-by-n matrix A is said to be totally nonnegative if every minor of A is nonnegative. Our main interest lies in characterizing all the inequalities that exist among products of principal minors of totally nonnegative matrices. We provide a complete description of all such inequalities for n-by-n totally nonnegative matrices with n&les;5. Other general results are also proved including a characterization when detA[α1]detA[α2]&les;detA[β1]detA[β2], for index sets α1,α2,β1,β2, and A totally nonnegative, along with various set-theoretic operations that preserve inequalities with respect to the totally nonnegative matrices. Many aspects of bidiagonal factorizations for totally nonnegative matrices are employed throughout. [Copyright &y& Elsevier]

We consider the class of the totally nonnegative matrices, i.e., the matrices having all their minors nonnegative, and intervals of matrices with respect to the chequerboard partial ordering, which results from the usual entrywise partial ordering if we reverse the inequality sign in all components having odd index sum. For these intervals in 1982 we stated in this journal the following conjecture: If the left and right endpoints of an interval are nonsingular and totally nonnegative, then all matrices taken from the interval are nonsingular and totally nonnegative. In this paper we show that this conjecture is true if we restrict ourselves to the subset of the almost totally positive matrices. [Copyright &y& Elsevier]

An n-by-n matrix is called totally nonnegative if every minor of A is nonnegative. The problem of interest is to characterize all possible Jordan canonical forms (Jordan structures) of irreducible totally nonnegative matrices. We show that the positive eigenvalues of such matrices have algebraic multiplicity one, and also demonstrate key relationships between the number and sizes of the Jordan blocks corresponding to zero. These notions yield a complete description of all Jordan forms through n = 7, as well as numerous general results. We also define a notion of ‘principal rank’ and employ this idea throughout. [ABSTRACT FROM AUTHOR]

A matrix A is called totally positive (TP) if all its minors are positive, and totally nonnegative (TN) if all its minors are nonnegative. A square matrix A is called oscillatory if it is TN and some power of A is TP. A linear time-varying system is called an oscillatory discrete-time system (ODTS) if the matrix defining its evolution at each time k is oscillatory. We analyze the properties of n -dimensional time-varying nonlinear discrete-time systems whose variational system is an ODTS, and show that they have a well-ordered behavior. More precisely, if the nonlinear system is time-varying and T -periodic then any trajectory either leaves any compact set or converges to an ((n − 1) T) -periodic trajectory, that is, a subharmonic trajectory. These results hold for any dimension n. The analysis of such systems requires establishing that a line integral of the Jacobian of the nonlinear system is an oscillatory matrix. This is non-trivial, as the sum of two oscillatory matrices is not necessarily oscillatory, and this carries over to integrals. We derive several new sufficient conditions guaranteeing that the line integral of a matrix is oscillatory, and demonstrate how this yields interesting classes of discrete-time nonlinear systems that admit a well-ordered behavior. [ABSTRACT FROM AUTHOR]