Lower Triangular Toeplitz Matrix Research Articles

The Cyclic Groups Generated by Catalan Matrices

Let $n$ a be positive integer. The Catalan matrix of order $n$, denoted by $\mathscr{C}_n[x]$, is a lower triangular Toeplitz matrix whose nonzero elements are expressions containing Catalan numbers and a real parameter $x$. In this paper, we investigate the multiplicative order of the cyclic group generated by $\mathscr{C}_n[x]$ (for $x\in\mathbb{N}$) modulo a positive integer $m$.

The smallest singular value of certain Toeplitz-related parametric triangular matrices

Abstract Let L be the infinite lower triangular Toeplitz matrix with first column (µ, a 1, a 2, ..., ap , a 1, ..., ap , ...) T and let D be the infinite diagonal matrix whose entries are 1, 2, 3, . . . Let A := L + D be the sum of these two matrices. Bünger and Rump have shown that if p = 2 and certain linear inequalities between the parameters µ, a 1, a 2, are satisfied, then the singular values of any finite left upper square submatrix of A can be bounded from below by an expression depending only on those parameters, but not on the matrix size. By extending parts of their reasoning, we show that a similar behaviour should be expected for arbitrary p and a much larger range of values for µ, a 1, ..., ap . It depends on the asymptotics in µ of the l 2-norm of certain sequences defined by linear recurrences, in which these parameters enter. We also consider the relevance of the results in a numerical analysis setting and moreover a few selected numerical experiments are presented in order to show that our bounds are accurate in practical computations.

Using q-calculus to study LDLt factorization of a certain Vandermonde matrix

We use tools from $q$-calculus to study $LDL^T$ decomposition of the Vandermonde matrix $V_q$ with coefficients $v_{i,j}=q^{ij}$. We prove that the matrix $L$ is given as a product of diagonal matrices and the lower triangular Toeplitz matrix $T_q$ with coefficients $t_{i,j}=1/(q;q)_{i-j}$, where $(z;q)_k$ is the q-Pochhammer symbol. We investigate some properties of the matrix $T_q$, in particular, we compute explicitly the inverse of this matrix.

A Petrov--Galerkin Spectral Method of Linear Complexity for Fractional Multiterm ODEs on the Half Line

We present a new tunably accurate Laguerre Petrov--Galerkin spectral method for solving linear multiterm fractional initial value problems with derivative orders at most one and constant coefficients on the half line. Our method results in a matrix equation of special structure which can be solved in $\mathcal{O}(N \log N)$ operations. We also take advantage of recurrence relations for the generalized associated Laguerre functions (GALFs) in order to derive explicit expressions for the entries of the stiffness and mass matrices, which can be factored into the product of a diagonal matrix and a lower-triangular Toeplitz matrix. The resulting spectral method is efficient for solving multiterm fractional differential equations with arbitrarily many terms, which we demonstrate by solving a fifty-term example. We apply this method to a distributed order differential equation, which is approximated by linear multiterm equations through the Gauss--Legendre quadrature rule. We provide numerical examples demonstra...

Continuous analogs of Schur extension problems and bitangential generalizations

The classical Schur extension problem is to describe the set of contractive holomorphic functions s(ζ) in the open unit disk of the form It is well-known that this set is nonempty if and only if the (n + 1) × (n + 1) lower triangular Toeplitz matrix A with entries aij = ai − j is contractive, i.e., if and only if In + 1 − AA* ≥ 0; see e.g., p. 79 of 5. In this paper the analogue of this problem for mvf's (matrix valued functions) that are holomorphic and contractive in the open upper half plane and bitangential generalizations thereof are studied.

Ballot matrix as Catalan matrix power and related identities

We use an analytical approach to find the k th 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.

Inversion of the generalized Fibonacci matrix by convolution

A convolution formula containing the generalized Fibonacci numbers and applications of this formula are investigated. Starting from the convolution formula, we derive combinatorial identities involving generalized and usual Fibonacci numbers, as well as the Lucas numbers. The inversion of a lower triangular matrix and the generalized inversion of strictly lower triangular Toeplitz matrix whose non-zero elements are generalized Fibonacci numbers are considered.

Inverting linear combinations of identity and generalized Catalan matrices

We introduce the notion of the generalized Catalan matrix as a kind of lower triangular Toeplitz matrix whose nonzero elements involve the generalized Catalan numbers. Inverse of the linear combination of the Pascal matrix with the identity matrix is computed in Aggarwala and Lamoureux (2002) [1]. In this paper, continuing this idea, we invert various linear combinations of the generalized Catalan matrix with the identity matrix. A simple and efficient approach to invert the Pascal matrix plus one in terms of the Hadamard product of the Pascal matrix and appropriate lower triangular Toeplitz matrices is considered in Yang and Liu (2006) [14]. We derive representations for inverses of linear combinations of the generalized Catalan matrix and the identity matrix, in terms of the Hadamard product which includes the Generalized Catalan matrix and appropriate lower triangular Toeplitz matrix.

Catalan matrix and related combinatorial identities

We introduce the notion of the Catalan matrix C n [ x ] whose non-zero elements are expressions which contain the Catalan numbers arranged into a lower triangular Toeplitz matrix. Inverse of the Catalan matrix is derived. Correlations between the matrix C n [ x ] and the generalized Pascal matrix are considered. Some combinatorial identities involving Catalan numbers, binomial coefficients and the generalized hypergeometric function are derived using these correlations. Moreover, an additional explicit representation of the Catalan number, as well as an explicit representation of the sum of the first m Catalan numbers are given.

A Bound for the Inverse of a Lower Triangular Toeplitz Matrix

The expression of a bound of the uniform norm of infinite lower triangular Toeplitz matrices with nonnegative entries is found. All the results are obtained by studying the behavior of the resolvent kernel and of the fundamental matrix of the recurrence relation, which generates the sequence of the entries of the considered matrix.

Unfalsified model parametrization based on frequency domain noise information

In this paper, the results concerning unfalsified model parametrization are extended to the case where identification noises are magnitude bounded in the frequency domain. It is shown that while every unfalsified model can still be represented by a linear fractional transformation of a block diagonal norm bounded transfer function matrix, the dimensions of the involved matrix valued transfer functions are of the order of the identification experiment data length. Moreover, the verification of the parametrization condition only requires computing the maximal singular values of two constant matrices. These results are established on the basis of a relation between a lower-triangular Toeplitz matrix and the discrete Fourier transformation of its first column elements. However, an analysis shows that the parametrization condition can be satisfied only when the noise level is very low. This implies that the proposed least-squares minimization-based algorithms are not very appropriate in model set identification.