On matrix equivalence and matrix equations

Roth's theorem on the solvability of matrix equations of the form AX?YB=C is proved for matrices with coefficients from a division ring or a ring which is module-finite over its center.     

Identities via Bell matrix and Fibonacci matrix

In this paper, we study the relations between the Bell matrix and the Fibonacci matrix, which provide a unified approach to some lower triangular matrices, such as the Stirling matrices of both kinds, the Lah matrix, and the generalized Pascal matrix. To make the results more general, the discussion is also extended to the generalized Fibonacci numbers and the corresponding matrix. Moreover, based on the matrix representations, various identities are derived.     

The chainable matrix,a special combinatorial matrix

The results of this paper comprise a study of a special class of combinatorial matrices called chainable matrices. The structure and algebraic behavior of these matrices are given specific attention.     

ON HERMITE MATRIX POLYNOMIALS AND HERMITE MATRIX FUNCTIONS

In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.     

Computing the matrix exponential and other matrix functions

In this note we present an algorithm for the computation of matrix functions, especially the matrix exponential. It is a new application of the so-called ‘elimination method’, which was presented for the scalar case in previous papers (e.g. [5]).     

On a matrix associated to a matrix pair

For a c–stable matrix pair (E,A), i.e., all finite eigenvalues of (E,A) have negative real part, an associated c–stable matrix is provided. It has all finite eigenvalues of (E,A) as eigenvalues and an additional stable eigenvalue −α, where α may be chosen arbitrarily, which corresponds to the eigenvalue ∞ of (E,A). In the case of positive systems, this associated c–stable matrix is, in addition, shown to be a −M-matrix. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Addition requirements for matrix and transposed matrix products

Let M be an s × t matrix and let M^T be the transpose of M. Let x and y be t- and s-dimensional indeterminate column vectors, respectively. We show that any linear algorithm A that computes Mx has associated with it a natural dual linear algorithm denoted A^T that computes M^Ty. Furthermore, if M has no zero rows or columns then the number of additions used by A^T exceeds the number of additions used by A by exactly s − t. In addition, a strong correspondence is established between linear algorithms that compute the product Mx and bilinear algorithms that compute the bilinear form y^TMx.     

Radicals of matrix rings generated by a companion matrix

The radicals of full matrix rings as well as structural matrix rings have been studied extensively and there is a well-developed theory for both. Here we initiate the radical theory for another class of matrix rings over a given ring; the matrix ring generated by the companion matrix of a polynomial over the ring.     

On-line soft error correction in matrix–matrix multiplication

Soft errors are one-time events that corrupt the state of a computing system but not its overall functionality. Soft errors normally do not interrupt the execution of the affected program, but the affected computation results cannot be trusted any more. A well known technique to correct soft errors in matrix–matrix multiplication is algorithm-based fault tolerance (ABFT). While ABFT achieves much better efficiency than triple modular redundancy (TMR) – a traditional general technique to correct soft errors, both ABFT and TMR detect errors off-line after the computation is finished. This paper extends the traditional ABFT technique from off-line to on-line so that soft errors in matrix–matrix multiplication can be detected in the middle of the computation during the program execution and higher efficiency can be achieved by correcting the corrupted computations in a timely manner. Experimental results demonstrate that the proposed technique can correct one error every ten seconds with negligible (i.e. less than 1%) performance penalty over the ATLAS dgemm().     

Schröder matrix as inverse of Delannoy matrix

Using Riordan arrays, we introduce a generalized Delannoy matrix by weighted Delannoy numbers. It turns out that Delannoy matrix, Pascal matrix, and Fibonacci matrix are all special cases of the generalized Delannoy matrices, meanwhile Schröder matrix and Catalan matrix also arise in involving inverses of the generalized Delannoy matrices. These connections are the focus of our paper. The half of generalized Delannoy matrix is also considered. In addition, we obtain a combinatorial interpretation for the generalized Fibonacci numbers.     

Multivariate matrix refinable functions with arbitrary matrix dilation

Characterizations of the stability and orthonormality of a multivariate matrix refinable function with arbitrary matrix dilation are provided in terms of the eigenvalue and -eigenvector properties of the restricted transition operator. Under mild conditions, it is shown that the approximation order of is equivalent to the order of the vanishing moment conditions of the matrix refinement mask . The restricted transition operator associated with the matrix refinement mask is represented by a finite matrix , with and being the Kronecker product of matrices and . The spectral properties of the transition operator are studied. The Sobolev regularity estimate of a matrix refinable function is given in terms of the spectral radius of the restricted transition operator to an invariant subspace. This estimate is analyzed in an example.

     

The sign matrix and the separation of matrix eigenvalues

The sign matrices uniquely associated with the matrices (M ? ζ_jI)², where ζ_j are the corners of a rectangle oriented at π/4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.     

Hermite matrix polynomials and second order matrix differential equations

In this paper we introduce the class of Hermite’s matrix polynomials which appear as finite series solutions of second order matrix differential equations Y″−xAY′+BY=0. An explicit expression for the Hermite matrix polynomials, the orthogonality property and a Rodrigues’ formula are given.     

On the best approximation matrix problem and matrix Fourier series

In this paper the concept of positive definite bilinear matrix moment functional, acting on the space of all the matrix valued continuous functions defined on a bounded interval [a,b] is introduced. The best approximation matrix problem with respect to such a functional is solved in terms of matrix Fourier series. Basic properties of matrix Fourier series such as the Riemann—Lebesgue, matrix property and the bessel—parseval matrix inequality are proved. The concept of total set with respect to a positive definite matrix functional is introduced, and the totallity of an orthonormal sequence of matrix polynomials with respect to the functional is established.     

Generalized matrix inversion is not harder than matrix multiplication

Starting from the Strassen method for rapid matrix multiplication and inversion as well as from the recursive Cholesky factorization algorithm, we introduced a completely block recursive algorithm for generalized Cholesky factorization of a given symmetric, positive semi-definite matrix A∈R^n×n$A\in {\mathbb{R}}^{n\times n}$. We used the Strassen method for matrix inversion together with the recursive generalized Cholesky factorization method, and established an algorithm for computing generalized {2,3}$\{2,3\}$ and {2,4}$\{2,4\}$ inverses. Introduced algorithms are not harder than the matrix–matrix multiplication.     

Completion of a partial integral matrix to a unimodular matrix

We first characterize submatrices of a unimodular integral matrix. We then prove that if n entries of an n × n partial integral matrix are prescribed and these n entries do not constitute a row or a column, then this matrix can be completed to a unimodular matrix. Consequently an n × n partial integral matrix with n − 1 prescribed entries can always be completed to a unimodular matrix.     

Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the kth power of the Catalan matrix. Precisely, it is proven that the power of the Catalan matrix is a lower triangular Toeplitz matrix which contains the well-known ballot numbers. A result from [H. S. Wilf, Generatingfunctionology, Academic Press, New York, 1990, Free download available from http://www.math.upenn.edu/~wilf/Downld.html.], related to the generating function for Catalan numbers, is extended to the negative integers. Three interesting representations for Catalan numbers by means of the binomial coefficients and the hypergeometric functions are obtained using relations between Catalan matrix powers.     

Rectangular matrix Padé approximants and square matrix orthogonal polynomials

In this paper we study Padé-type and Padé approximants for rectangular matrix formal power series, as well as the formal orthogonal polynomials which are a consequence of the definition of these matrix Padé approximants. Recurrence relations are given along a diagonal or two adjacent diagonals of the table of orthogonal polynomials and their adjacent ones. A matrix qd-algorithm is deduced from these relations. Recurrence relations are also proved for the associated polynomials. Finally a short presentation of right matrix Padé approximants gives a link between the degrees of orthogonal polynomials in right and left matrix Padé approximants in order to show that the latter are identical. This revised version was published online in June 2006 with corrections to the Cover Date.     

Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.     

Convergent matrix pencils

This paper deals with the study of sufficient conditions to guarantee that matrices expressed as a matrix pencil of a pair of given matrices be convergent.