Bounds on the inverses of nonnegative triangular matrices with monotonicity properties

Some bounds on the entries and on the norm of the inverse of triangular matrices with nonnegative and monotone entries are found. All the results are obtained by exploiting the properties of the fundamental matrix of the recurrence relation which generates the sequence of the entries of the inverse matrix. One of the results generalizes a theorem contained in a recent article of one of the authors about Toeplitz matrices.     

An optimal bound for inverses of triangular matrices with monotone entries

This article provides a new bound for 1-norms of inverses of positive triangular matrices with monotonic column entries. The main theorem refines a recent inequality established in Vecchio and Mallik [Bounds on the inverses of non-negative lower triangular Toeplitz matrices with monotonicity properties, Linear Multilinear Alg., 55 (2007), pp. 365–379]. The results are shown to be in a sense best possible under the given constraints.     

Cauchy型矩阵及其逆矩阵的快速三角分解

Abstract, In this paper,algorithms for determining the triangular factorization of Cauchy typematrices and their inverses are derived by using O(n2) operations.     

Preserving diagonalisability on upper triangular matrices

We characterise all bijective linear mappings on the algebra of upper triangular _n × n matrices that preserve diagonalisability.     

The index of triangular operator matrices

For any triangular operator matrix acting in a direct sum of complex Banach spaces, the order of a pole of the resolvent (i.e. the index) is determined as a function of the coefficients in the Laurent series for all the (resolvents of the) operators on the diagonal and of the operators below the diagonal. This result is then applied to the case of certain nonnegative operators in Banach lattices. We show how simply these results imply the Rothblum Index Theorem (1975) for nonnegative matrices. Finally, examples for calculating the index are presented.

     

On the determinants and inverses of circulant matrices with Fibonacci and Lucas numbers

Let A_n=Circ(F₁,F₂,…,F_n) and B_n=Circ(L₁,L₂,…,L_n) be circulant matrices, where F_n is the Fibonacci number and L_n is the Lucas number. We prove that A_n is invertible for n > 2, and B_n is invertible for any positive integer n. Afterwards, the values of the determinants of matrices A_n and B_n can be expressed by utilizing only the Fibonacci and Lucas numbers. In addition, the inverses of matrices A_n and B_n are derived.     

On weak monotonicity of the powers of nonnegative matrices

The purpose of this paper is to study conditions under which the powers of a square matrix A are weak monotone. Necessary and sufficient conditions under which the product of any two weak monotone matrices is weak monotone are also obtained. Several examples are given to illustrate the necessity of the conditions stated.     

On constrained generalized inverses of matrices and their properties

A matrix G is called a generalized inverse (g-invserse) of matrix A if AGA = A and is denoted by G = A ⁻. Constrained g-inverses of A are defined through some matrix expressions like E(AE)⁻, (FA)⁻ F and E(FAE)⁻ F. In this paper, we derive a variety of properties of these constrained g-inverses by making use of the matrix rank method. As applications, we give some results on g-inverses of block matrices, and weighted least-squares estimators for the general linear model.     

A dominance notion for singular matrices with applications to nonnegative generalized inverses

A dominance rule for singular matrices using proper splittings is proposed. This extends the corresponding notion, known for nonsingular matrices. An application to the nonnegativity of the Moore–Penrose inverse is presented.     

A note on representations for the inverses of tridiagonal matrices

In this note, we propose an explicit representation with the nested sums for the entries of the inverses of general tridiagonal nonsingular matrices. Its equivalence with other particular representations, based on the combinatorial expressions or the continued fractions, is considered. In addition, an analytical representation for the entries of the finite sections of the resolvent of Jacobi matrices, in terms of its related orthogonal polynomials, is observed.     

Normal dilatation of triangular matrices

LetR be a (real or complex) triangular matrix of ordern, say, an upper triangular matrix. Is it true that there exists a normaln×n matrixA whose upper triangle coincides with the upper triangle ofR? The answer to this question is “yes” and is obvious in the following cases: (1)R is real; (2)R is a complex matrix with a real or a pure imaginary main diagonal, and moreover, all the diagonal entries ofR belong to a straight line. The answer is also in the affirmative (although it is not so obvious) for any matrixR of order 2. However, even forn=3 this problem remains unsolved. In this paper it is shown that the answer is in the affirmative also for 3×3 matrices.     

A monotonicity formula on complete Kähler manifolds with nonnegative bisectional curvature

In this paper, we derive a new monotonicity formula for the plurisubharmonic functions/positive (1,1) currents on complete Kähler manifolds with nonnegative bisectional curvature. As applications we derive the sharp estimates for the dimension of the spaces of holomorphic functions (sections) with polynomial growth, which, in particular, partially solve a conjecture of Yau.

The methods used in this paper, without the assumption of maximum volume of growth, as observed recently by Chen, Fu, Yin, and Zhu, provide a complete solution to Yau's conjecture.

     

On the eigenvalues of matrices with prescribed upper triangular part

Consider a matrix D=[D_i,j] partitioned in submatrices D_i,j where the submatrices D_i,iare square. This paper studies the possible eigenvalues of D, when the blocks D_i,j, with i≤j, are fixed, and the other blocks vary. The result obtained generalizes twotheorems already known.     

An efficient method for computing the inverse of arrowhead matrices

In this paper we propose a simple and effective method to find the inverse of arrowhead matrices which often appear in wide areas of applied science and engineering such as wireless communications systems, molecular physics, oscillators vibrationally coupled with Fermi liquid, and eigenvalue problems. A modified Sherman–Morrison inverse matrix method is proposed for computing the inverse of an arrowhead matrix. The effectiveness of the proposed method is illustrated and numerical results are presented along with comparative results.     

The representations of generalized inverses of lower triangular operators

In this note, the explicit representations of Moore-Penrose inverses of lower triangular operator matrices are established. As an application, the explicit representations of Bott-Duffin inverses of bounded linear operators on a Hilbert space with respect to a closed subspace are obtained.     

Possible numbers of positive entries of imprimitive nonnegative matrices

Zhan, X., Extremal numbers of positive entries of imprimitive nonnegative matrix, Linear Algebra Appl. (in press) has determined the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices with a given imprimitivity index. Let σ( A ) denote the number of positive entries of a matrix A. Let M(n,?k) and m(n,?k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In this article, we prove that for any positive integer d with m(n,k)≤ d?≤?M(n,k), there exists an n?×?n irreducible nonnegative matrix A with imprimitivity index k such that?σ?(A)=d.     

Sign patterns for eigenmatrices of nonnegative matrices

For a square (0,?1,??1) sign pattern matrix S, denote the qualitative class of S by Q(S). In this article, we investigate the relationship between sign patterns and matrices that diagonalize an irreducible nonnegative matrix. We explicitly describe the sign patterns S such that every matrix in Q(S) diagonalizes some irreducible nonnegative matrix. Further, we characterize the sign patterns S such that some member of Q(S) diagonalizes an irreducible nonnegative matrix. Finally, we provide necessary and sufficient conditions for a multiset of real numbers to be realized as the spectrum of an irreducible nonnegative matrix M that is diagonalized by a matrix in the qualitative class of some S ² NS sign pattern.