Positive diagonal solutions to the Lyapunov equations

We study various stability type conditions on a matrix A related to the consistency of the Lyapunov equation AD+DA^t positive definite, where D is a positive diagonal matrix. Such problems arise in mathematical economics, in the study of time-invariant continuous-time systems and in the study of predator-prey systems. Using a theorem of the alternative, a characterization is given for all A satisfying the above equation. In addition, some necessary conditions for consistency and some related ideas are discussed. Finally, a method for constructing a solution D to the equation is given for matrices A satisfying certain conditions.     

Geršgorin variations III: On a theme of Brualdi and Varga

Brualdi brought to Geršgorin Theory the concept that the digraph G(A) of a matrix A is important in studying whether A is singular. He proved, for example, that if, for every directed cycle of G(A), the product of the diagonal entries exceeds the product of the row sums of the moduli of the off-diagonal entries, then the matrix is nonsingular. We will show how, in polynomial time, that condition can be tested and (if satisfied) produce a diagonal matrix D, with positive diagonal entries, such that AD (where A is any nonnnegative matrix satisfying the conditions) is strictly diagonally dominant (and so, A is nonsingular). The same D works for all matrices satisfying the conditions. Varga raised the question of whether Brualdi’s conditions are sharp. Improving Varga’s results, we show, if G is scwaltcy (strongly connected with at least two cycles), and if the Brualdi conditions do not hold, how to construct (again in polynomial time) a complex matrix whose moduli satisfy the given specifications, but is singular.     

Drazin inverse of partitioned matrices in terms of Banachiewicz-Schur forms

Let be a partitioned matrix, where A and D are square matrices. Denote the Drazin inverse of A by A^D. The purpose of this paper is twofold. Firstly, we develop conditions under which the Drazin inverse of M having generalized Schur complement, S=D-CA^DB, group invertible, can be expressed in terms of a matrix in the Banachiewicz-Schur form and its powers. Secondly, we deal with partitioned matrices satisfying rank(M)=rank(A^D)+rank(S^D), and give conditions under which the group inverse of M exists and a formula for its computation.     

On a class of matrices which arise in the numerical solution of Euler equations

Summary We study block matricesA=[A_ij], where every blockA _ij ^k,k is Hermitian andA _ii is positive definite. We call such a matrix a generalized H-matrix if its block comparison matrix is a generalized M-matrix. These matrices arise in the numerical solution of Euler equations in fluid flow computations and in the study of invariant tori of dynamical systems. We discuss properties of these matrices and we give some equivalent conditions for a matrix to be a generalized H-matrix.Research supported by the Graduiertenkolleg mathematik der Universität Bielefeld     

Affine minimal hypersurfaces of rotation

We give a complete list of affine minimal surfaces inA ³ with Euclidean rotational symmetry, completing the treatise given in [1] and prove that these surfaces have maximal affine surface area within the class of all affine surfaces of rotation satisfying suitable boundary conditions. Besides we show that for rotationally symmetric locally strongly convex affine minimal hypersurfaces inA ⁿ ,n4, the second variation of the affine surface area is negative definite under certain conditions on the meridian.     

Perturbation bounds for theLDL H andLU decompositions

LetA andA+A be Hermitian positive definite matrices. Suppose thatA=LDL ^H and (A+A)=(L+L)(D+D)(L+L)^H are theLDL ^H decompositons ofA andA+A, respectively. In this paper upper bounds on |D| _F and |L| _F are presented. Moreover, perturbation bounds are given for theLU decomposition of a complexn ×n matrix.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

The Constrained Solutions of Two Matrix Equations

We study the symmetric positive semidefinite solution of the matrix equation AX ₁ A ^T + BX ₂ B ^T = C, where A is a given real m×n matrix, B is a given real m×p matrix, and C is a given real m×m matric, with m, n, p positive integers; and the bisymmetric positive semidefinite solution of the matrix equation D ^T XD = C, where D is a given real n×m matrix, C is a given real m×m matrix, with m, n positive integers. By making use of the generalized singular value decomposition, we derive general analytic formulae, and present necessary and sufficient conditions for guaranteeing the existence of these solutions. Received December 17, 1999, Revised January 10, 2001, Accepted March 5, 2001     

The Picard iteration and its application

The convergence behavior of the Picard iteration X_k+1=AX_k+B and the weighted case Y_k=X_k/b_k is investigated. It is shown that the convergence of both these iterations is related to the so-called effective spectrum of A with respect to some matrix. As an application of our convergence results we discuss the convergence behavior of a sequence of scaled triangular matrices {D^NT^N }.     

Equalities and inequalities for inertias of hermitian matrices with applications

The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of 2×2 block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression A-BXB^∗ with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation AX=B, as well as the extremal inertias of a partial block Hermitian matrix.     

Uniform and minimal {0,1} – cp matrices

Let A be an n?×?n real matrix. A is called {0,1}-cp if it can be factorized as A?=?BB ^T with b_ij =0 or 1. The smallest possible number of columns of B in such a factorization is called the {0,1}-rank of A. A {0,1}-cp matrix A is called minimal if for every nonzero nonnegative n?×?n diagonal matrix D, A-D is not {0,1}-cp, and r-uniform if it can be factorized as A=BB ^T, where B is a (0,?1) matrix with r 1s in each column. In this article, we first present a necessary condition for a nonsingular matrix to be {0,1}-cp. Then we characterize r-uniform {0,1}-cp matrices. We also obtain some necessary conditions and sufficient conditions for a matrix to be minimal {0,1}-cp, and present some bounds for {0,1}-ranks.     

When the positivity of the leading principal minors implies the positivity of all principal minors of a matrix

An n-by-n real matrix A enjoys the “leading implies all” (LIA) property, if, whenever D   is a diagonal matrix such that A+D$A+D$ has positive leading principal minors (PMs), all PMs of A are positive. Symmetric and Z-matrices are known to have this property. We give a new class of matrices (“mixed matrices”) that both unifies and generalizes these two classes and their special diagonal equivalences by also having the LIA property. “Nested implies all” (NIA) is also enjoyed by this new class.     

Pentadiagonal Oscillatory Matrices with Two Spectrum in Common

We denote the spectrum of an square matrix A by σ(A), and that of the matrix obtained by deleting the first i rows and columns of A by σ_i(A). It is known that a symmetric pentadiagonal oscillatory (SPO) matrix may be constructed from σ, σ₁ and σ₂. The pairs σ, σ₁ and σ₁, σ₂ must interlace; the construction is not unique; and the conditions on the data which ensure that A is oscillatory are extremely complicated. Given one SPO matrix A, the paper shows that operations may be applied to A to construct a family of such matrices with σ and σ₁ in common. Moreover, given one totally positive (TP) matrix A, we construct a family of TP matrices with σ, σ₁ and σ₂ in common.     

Quadratic Lyapunov functions for systems with state-dependent switching

In this paper, we consider the existence of quadratic Lyapunov functions for certain types of switched linear systems. Given a partition of the state-space, a set of matrices (linear dynamics), and a matrix-valued function A(x) constructed by associating these matrices with regions of the state-space in a manner governed by the partition, we ask whether there exists a positive definite symmetric matrix P such that A(x)^TP+PA(x) is negative definite for all x(t). For planar systems, necessary and sufficient conditions are given. Extensions for higher order systems are also presented.     

Continuity of The Drazin inverse

Let A be an n×n matrix. It is shown that if a matrix  comes close to satisfying the definition of the Drazin inverse of A,A^D , then  is close to A^D .     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

On visualization scaling, subeigenvectors and Kleene stars in max algebra

The purpose of this paper is to investigate the interplay arising between max algebra, convexity and scaling problems. The latter, which have been studied in nonnegative matrix theory, are strongly related to max algebra. One problem is that of strict visualization scaling, defined as, for a given nonnegative matrix A, a diagonal matrix X such that all elements of X^-1AX are less than or equal to the maximum cycle geometric mean of A, with strict inequality for the entries which do not lie on critical cycles. In this paper such scalings are described by means of the max algebraic subeigenvectors and Kleene stars of nonnegative matrices as well as by some concepts of convex geometry.     

Positivity Preserving Hadamard Matrix Functions

For every positive real number p that lies between even integers 2(m − 2) and 2(m − 1) we demonstrate a matrix A = [a_ij] of order 2m such that A is positive definite but the matrix with entries |a_ij|^p is not.     

A note on Potter’s theorem for quasi-commutative matrices

We discuss the converse of a theorem of Potter stating that if the matrix equation AB=ωBA is satisfied with ω a primitive qth root of unity, then A^q+B^q=(A+B)^q. We show that both conditions have to be modified to get a converse statement and we present a characterization when the converse holds for these modified conditions and q=3 and a conjecture for the general case. We also present some further partial results and conjectures.     

由Smith标准形求解矩阵方程∑AiXBi=C

By using the Smith normal form of polynomial matrix and algebraic methods, this paper discusses the solvability for the linear matrix equation ∑AiXBi = C over a field, and obtains the explicit formulas of general solution or unique solution.