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This study associates the symmetric Bernoulli matrix with various combinatorial matrices, such as Pascal, Vandermonde, and Stirling matrices of the first and second kind including Catalan numbers by way of dual matrices of the matrix. A recurrence relation related to the entries of the Cholesky factor of a symmetric positive definite matrix is presented, which sheds new light on examining positive definiteness and Cholesky's method of a symmetric matrix. This relation is applied to give a necessary and sufficient condition for the positive definiteness of a symmetric matrix, which leads to a slightly modified form of the outer-product formulation for Cholesky's method of a symmetric matrix. [ABSTRACT FROM AUTHOR]

As an inverse relation, involution with an invariant sequence plays a key role in combinatorics and features prominently in some of Shapiro’s open questions (Shapiro, 2001). In this paper, invariant sequences are used to provide answers to some of these questions about the Fibonacci matrix and Riordan involutions. [ABSTRACT FROM AUTHOR]

We introduce the notion of dual matrices of an infinite matrix A , which are defined by the dual sequences of the rows of A and naturally connected to the Pascal matrix P = [ ( i j ) ] ( i , j = 0 , 1 , 2 , … ) . We present the Cholesky decomposition of the symmetric Pascal matrix by means of its dual matrix. Decompositions of a Vandermonde matrix are used to obtain variants of the Lagrange interpolation polynomial of degree ≤ n that passes through the n + 1 points ( i , q i ) for i = 0 , 1 , … , n . [ABSTRACT FROM AUTHOR]

An infinite real sequence { a n } is called an invariant sequence of the first (resp., second) kind if a n = ∑ k = 0 n ( n k ) ( − 1 ) k a k (resp., a n = ∑ k = n ∞ ( k n ) ( − 1 ) k a k ). We review and investigate invariant sequences of the first and second kind, and study their relationships using similarities of Pascal-type matrices and their eigenspaces. [ABSTRACT FROM AUTHOR]

Two vectors with the same number of components are called conformal if each pair of corresponding components has a nonnegative product. A sign-central matrix A is called tight sign-central if any two columns of A are not conformal. An m × n matrix A with n ≥2 is called conformal if any two columns of A are conformal. A submatrix M of an m × n matrix A is called a maximal conformal block (MCB) of A provided that M is conformal with m rows and any submatrices of A obtained from M by adding a column of A not in M are not conformal. A sign-central matrix A is called nearly tight sign-central (NTSC) if A has only one MCB. An NTSC matrix A is called minimal NTSC (MNTSC) if A is minimal sign-central. In this article, we investigate some combinatorial properties of NTSC matrices. We also prove that, for a positive integer m with m >2, there exists an m × n MNTSC matrix A if and only if m + 1 ≤n≥2m -- 1. Furthermore, we characterize an m × (2m -- 1) MNTSC matrix for each positive integer m > 2. [ABSTRACT FROM AUTHOR]

Abstract: Let denote a set of all elements of weighted lattice paths with weight in the xy-plane from to such that a vertical step , a horizontal step , and a diagonal step are endowed with weights , and respectively and let denote the weight of defined bywhere is the product of the weights of all its steps in . A matrix is called a lattice path matrix with weight if for a triple , and of real numbers . In this paper, we present LDU decomposition of lattice path matrices with weight and related properties for every triple , and of real numbers, and a necessary and sufficient condition in which the symmetric lattice path matrices are positive definite. We also investigate the relationship between the lattice path matrices and generalized Pascal matrices. [Copyright &y& Elsevier]

Abstract: A real matrix A is called sign-central if has a nonzero nonnegative solution x for every matrix with the same sign pattern as A. A sign-central matrix A is called tight sign-central if the Hadamard(entrywise) product of any two columns of A contains a negative component. Hwang et al. [S.G. Hwang, I.P. Kim, S.J. Kim, X.D. Zhang, Tight sign-central matrices, Linear Algebra Appl. 371 (2003) 225–240] proved that, for a positive integer m, there exists an m × n (0,1,−1) tight sign-central matrix A with no zero rows if and only if m +1⩽ n ⩽2 m . They also determined the lower bound of the number of columns of a tight sign-central matrix with no zero rows in terms of the number of rows and the number of zero entries of the matrix along with the characterization of the equality case. For an m × n matrix A, the sparsity of A is the ratio σ(A)/mn where σ(A) denotes the number of zero entries of A. In this paper, we determine the maximum number and the minimum number of zero entries of an m × n tight sign-central matrix with no zero rows for each pair (m, n) of positive integers with m +1⩽n⩽2 m . We also determine the maximum sparsity of tight sign-central matrices with m nonzero rows in terms of m for each positive integer m. [Copyright &y& Elsevier]

A real matrix A is called sign-central if the convex hull of the columns of A contains the zero vector 0 for every matrix A with the same sign pattern as A. A sign-central matrix A is called a minimal sign-central matrix if the deletion of any of the columns of A breaks the sign-centrality of A. A sign-central matrix A is called tight sign-central if the Hadamard (entrywise) product of any two columns of A contains a negative component. In this paper, we show that every tight sign-central matrix is minimal sign-central and characterize the tight sign-central matrices. We also determine the lower bound of the number of columns of a tight sign-central matrix in terms of the number of rows and the number of zero entries of the matrix. [Copyright &y& Elsevier]

This paper proposes an effective model predictive control (MPC) scheme using a modulated optimal vector (MOV) and finite control options for a three-phase inverter with an LC filter. Unlike other MPC methods, the proposed MPC strategy exploits the unconstrained optimal vector (OV) of the continuous-control-set (CCS) MPC to limit the control options for the unconstrained mode. First, the analytical OV is derived based on a least-squares optimization. If the input constraints are not violated, the OV is applied with a space vector modulation (SVM) technique like the CCS-MPC. Otherwise, the OV is scaled into the MOV and only three control options are online evaluated to reselect the control input. Experiments are conducted on a three-phase inverter test bed with a TI TMS320F28335 digital signal processor to validate the improvements of the proposed method, especially the robust performances and fast responses. The comparative results with the FCS-MPC show the superior performances of the proposed scheme with smaller steady-state error and lower total harmonic distortion due to the analytical OV with SVM, more robustness to parameter uncertainties due to the disturbance observer, and faster dynamic response due to the online reselection of control inputs. [ABSTRACT FROM PUBLISHER]

Abstract: Let and D=diag((−1)0,(−1)1,(−1)2,…). As a linear transformation of the infinite dimensional real vector space R∞={(x0,x1,x2,…)T ∣ xi ∈ R for all i}, PD has only two eigenvalues 1, −1. In this paper, we find some matrices associated with P whose columns form bases for the eigenspaces for PD. We also introduce truncated Fibonacci sequences and truncated Lucas sequences and show that these sequences span the eigenspaces of PD. [Copyright &y& Elsevier]