Nonzero Diagonal Entries Research Articles

Triangular intuitionistic fuzzy linear system of equations with applications: an analytical approach

This study extended an existing semi-analytical technique, the Homotopy Perturbation Method, to the Block Homotopy Modified Perturbation Method by solving two n × n crisp triangular intuitionistic fuzzy (TIF) systems of linear equations. In the original system, the coefficient matrix is considered as real crisp, while the unknown variable vector and right hand side vector are regarded as triangular intuitionistic fuzzy numbers. The Block Homotopy Modified Perturbation Method is found to be efficient and practical to solve n × n TIF linear systems as it only requires the non-singularity of the n × n TIF linear system's coefficient matrix, whereas the point Homotopy Perturbation Method and other classical numerical iterative methods typically require non-zero diagonal entries in the coefficient matrix. A set of theorems relevant to this study are presented and demonstrated. We solve an engineering application, i.e. a current flow circuit problem that is represented in terms of a triangular intuitionistic fuzzy environment, using the suggested method. The unknown current is then obtained as a triangle intuitionistic fuzzy number. The proposed semi-analytic method is used to solve some numerical test problems in order to validate their performance and efficiency in comparison to other existing techniques. The numerical results of the example are displayed on graphs with different degrees of uncertainty. The efficiency and accuracy of the proposed method are further demonstrated by comparisons to block Jacobi, Adomain Decomposition method, Successive Over-Relaxation method and the classical Gauss-Seidel numerical method.

Left-Invertibility of Rank-One Perturbations

For each isometry V acting on some Hilbert space and a pair of vectors f and g in the same Hilbert space, we associate a nonnegative number c(V; f, g) defined by $$\begin{aligned} c(V; f,g) = (\Vert f\Vert ^2 - \Vert V^*f\Vert ^2) \Vert g\Vert ^2 + |1 + \langle V^*f , g\rangle |^2. \end{aligned}$$We prove that the rank-one perturbation \(V + f \otimes g\) is left-invertible if and only if $$\begin{aligned} c(V;f,g) \ne 0. \end{aligned}$$We also consider examples of rank-one perturbations of isometries that are shift on some Hilbert space of analytic functions. Here, shift refers to the operator of multiplication by the coordinate function z. Finally, we examine \(D + f \otimes g\), where D is a diagonal operator with nonzero diagonal entries and f and g are vectors with nonzero Fourier coefficients. We prove that \(D + f\otimes g\) is left-invertible if and only if \(D+f\otimes g\) is invertible.

Hypercyclic shift factorizations for bilateral weighted shift operators

Taking the perspective that a bilateral weighted shift is an operator that shifts some two-sided canonical basic sequence of ℓp(Z), with 1⩽p<∞, we show that every bilateral weighted shift on ℓp(Z) has a factorization T=AB, where A and B are hypercyclic bilateral weighted shifts. For the case when T is invertible, both shifts A and B may be selected to be invertible as well. Moreover, we show analogous hypercyclic factorization results for diagonal operators with nonzero diagonal entries.

The minrank of random graphs over arbitrary fields

The minrank of a graph G on the set of vertices [n] over a field $$\mathbb{F}$$ is the minimum possible rank of a matrix $$M \in \mathbb{F}^{{n}\times{n}}$$ with nonzero diagonal entries such that Mi,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field $$\mathbb{F}=\mathbb{F}(n)$$ and every p = p(n) satisfying n-1 ≤ p ≤ 1 - n-0.99, the minrank of G = G(n, p) over $$\mathbb{F}$$ is $$\Theta(\frac{n {\rm{log}}(1/p)}{{\rm{log}} n})$$ with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev and Weinstein, who proved the above result for finite fields of size at most nO(1), with tools from linear algebra, including an estimate of Rónyai, Babai and Ganapathy for the number of zero-patterns of a sequence of polynomials.

Automorphism groups of some generalized Cayley graphs

Let R be a commutative ring with identity element. Graph \(\Gamma ^{n}_{R}\) is defined with vertex set \(R^{n} \setminus \lbrace 0 \rbrace \) and two distinct vertices X and Y are adjacent if and only if there exists an \(n \times n\) lower triangular matrix A with non-zero diagonal entries such that \(AX^T=Y^T\) or \(AY^T=X^T\). By \(B^{T}\), we mean transpose of matrix B. If R is a semigroup with respect to multiplication and \(n=1,\) then \(\Gamma ^{1}_{R}\) is the undirected Cayley graph. In this paper, a prime number p, we find the clique number and automorphism group of \(\Gamma ^{n}_{R},\) where \(R=\mathbb {Z}_{p^2}\) or \(R=\mathbb {Z}_{p^3}\).

Adaptive Fuzzy Sliding Mode Control of Under-actuated Nonlinear Systems

A new extension of the conventional adaptive fuzzy sliding mode control (AFSMC) scheme, for the case of under-actuated and uncertain affine multiple-input multiple-output (MIMO) systems, is presented. In particular, the assumption for non-zero diagonal entries of the input gain matrix of the plant is relaxed. In other words, the control effect of one actuator can propagate from a subgroup of canonical state equations to the rest of equations in an indirect sense. The asymptotic stability of the proposed AFSM control method is proved using a Lyapunov-based methodology. The effectiveness of the proposed method for the case of under-actuated systems is investigated in the presence of plant uncertainties and disturbances, through simulation studies.

A new characterization of simultaneous Lyapunov diagonal stability via Hadamard products

A well-known characterization by Kraaijevanger [14] for Lyapunov diagonal stability states that a real, square matrix A is Lyapunov diagonally stable if and only if A∘S is a P-matrix for any positive semidefinite S with nonzero diagonal entries. This result is extended here to a new characterization involving similar Hadamard multiplications for simultaneous Lyapunov diagonal stability on a set of matrices. Among the main ingredients for this extension are a new concept called P-sets and a recent result regarding simultaneous Lyapunov diagonal stability by Berman, Goldberg, and Shorten [2].

New Nonsingularity Conditions for General Matrices and the Associated Eigenvalue Inclusion Sets

The paper suggests generalizations of some known sufficient nonsingularity conditions for matrices with constant principal diagonal and the corresponding eigenvalue inclusion sets to the cases of arbitrary matrices and matrices with nonzero diagonal entries. Bibliography: 11 titles.

Measurement matrix optimization method based on matrix orthogonal similarity transformation

Optimization of the measurement matrix is one of the important research aspects of compressive sensing theory. A measurement matrix optimization method is presented based on the orthogonal similarity transformation of the information operator’s Gram matrix. In terms of the fact that the information operator’s Gram matrix is a singular symmetric matrix, a simplified orthogonal similarity transformation is deduced, and thus the simplified diagonal matrix that is orthogonally similar to it is obtained. Then an approximation of the Gram matrix is obtained by letting all the nonzero diagonal entries of the simplified diagonal matrix equal their average value. Thus an optimized measurement matrix can be acquired according to its relationship with the information operator. Results of experiments show that the optimized measurement matrix compared to the random measurement matrix is less coherent with dictionaries. The relative signal recovery error also declines when the proposed measurement matrix is utilized.

Representing polynomials as characteristic polynomials via the Stieltjes transform

The Stieltjes transform SA of an infinite lower triangular matrix A with nonzero diagonal entries is defined by SA=A−1A¯ where A¯ is the matrix obtained from A by deleting its initial row. In this paper, we express a sequence of polynomials as the characteristic polynomials of the Stieltjes transforms using a highly structured infinite lower triangular matrix called a Riordan matrix. As a result, computation of the zeros of such polynomials becomes amenable to iterative methods for computing eigenvalues, or to eigenvalue location theorems such as the Geršgorin theorem. We also describe a finite analog of the polynomial correspondence and its relationship to eigenvalue regions. As an application, the recurrence relations for several polynomial sequences are obtained using the Stieltjes transform.

The base of a primitive, nonpowerful sign pattern with exactly [formula omitted] nonzero diagonal entries

In [J.Y. Shao, L.H. You, H.Y. Shan, Bound on the base of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 2–3, 285–300], Shao, You and Shan extended the concept of the base from powerful sign pattern matrices to nonpowerful (generalized) sign pattern matrices. It is well known that the properties of the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base set of the primitive nonpowerful square sign pattern matrices of order n with exactly d (with d ≥ 1 ) nonzero diagonal entries. The base set is shown to be { 2 , 3 , … , 3 n − d − 1 } . The extremal sign pattern matrices with both the least number n + d nonzero entries and the maximum base 3 n − d − 1 are characterized.

A Graph-Theoretic Method for Choosing a Spanning Set for a Finite-Dimensional Vector Space, with Applications to the Grossman-Larson-Wright Module and the Jacobian Conjecture

It is well known that a square zero pattern matrix guarantees non-singularity if and only if it is permutationally equivalent to a triangular pattern with nonzero diagonal entries. It is also well known that a nonnegative square pattern matrix with positive main diagonal is sign nonsingular if and only if its associated digraph does not have any directed cycles of even length. Any $m\times n$ matrix containing an $n\times n$ sub-matrix with either of these forms will have full rank. We translate this idea into a graph-theoretic method for finding a spanning set of vectors for a finite-dimensional vector space from among a set of vectors generated combinatorially. This method is particularly useful when there is no convenient ordering of vectors and no upper bound to the dimensions of the vector spaces we are dealing with. We use our method to prove three properties of the Grossman-Larson-Wright module originally described by David Wright: $\overline{\cal M}(3,\infty)_m=0$ for $m\ge 3$, $\overline{\cal M}(4,3)_m=0$ for $5\le m\le 8$, and $\overline{\cal M}(4,4)_8=0$. The first two properties yield combinatorial proofs of special cases of the homogeneous symmetric reduction of the Jacobian conjecture.

Equal-diagonal QR decomposition and its application to precoder design for successive-cancellation detection

In multiple-input multiple-output (MIMO) multiuser detection theory, the QR decomposition of the channel matrix H can be used to form the back-cancellation detector. In this paper, we propose an optimal QR decomposition, which we call the equal-diagonal QR decomposition, or briefly the QRS decomposition. We apply the decomposition to precoded successive-cancellation detection, where we assume that both the transmitter and the receiver have perfect channel knowledge. We show that, for any channel matrix H, there exists a unitary precoder matrix S, such that HS=QR, where the nonzero diagonal entries of the upper triangular matrix R in the QR decomposition of HS are all equal to each other. The precoder and the resulting successive-cancellation detector have the following properties. a) The minimum Euclidean distance between two signal points at the channel output is equal to the minimum Euclidean distance between two constellation points at the precoder input up to a multiplicative factor that equals the diagonal entry in the R-factor. b) The superchannel HS naturally exhibits an optimally ordered column permutation, i.e., the optimal detection order for the vertical Bell Labs layered space-time (V-BLAST) detector is the natural order. c) The precoder S minimizes the block error probability of the QR successive cancellation detector. d) A lower and an upper bound for the free distance at the channel output is expressible in terms of the diagonal entries of the R-factor in the QR decomposition of a channel matrix. e) The precoder S maximizes the lower bound of the channel's free distance subject to a power constraint. f) For the optimal precoder S, the performance of the QR detector is asymptotically (at large signal-to-noise ratios (SNRs)) equivalent to that of the maximum-likelihood detector (MLD) that uses the same precoder. Further, We consider two multiplexing schemes: time-division multiple access (TDMA) and orthogonal frequency-division multiplexing (OFDM). We d

Bases of the Homology Spaces of the Hibert Scheme of Points in an Algebraic Surface

For a complex surface S , proper, smooth and connected, the authors find two bases of the spaces of rational homology H n (Hilb d S) Q of the Hilbert scheme of subschemes of S of length d . The idea of the proof of the main theorem is to prove that the elements of the two candidates have as cardinalities the known Betti numbers of Hilb d S and to show that both intersect in a triangular matrix of nonzero diagonal entries. Papers on the subject which have a close connection with the present one are by B. Fantechi [Base of the homology groups of the Hilbert scheme of points on a surface'', Preprint; per bibl.] and L. Gottsche [Math. Ann. 286 (1990), no. 1-3, 193–207 ].

Criteria for invertibility of diagonally dominant matrices

We develop some basic properties of finite diagonally dominant matrices. These properties are used to establish a necessary and sufficient condition for a finite diagonally dominant matrix with nonzero diagonal entries to be singular. This condition relates the nonstrict diagonally dominant rows of the matrix to the difference between the principal arguments of the nonzero entries along each column in these rows. The infinite dimensional case is also studied, where a sufficient condition for the invertibility of the matrix operator in the sequence space c 0 defined by a diagonally dominant infinite matrix A with nonzero diagonal entries is introduced. This sufficient condition improves some of the earlier results. When the sequence space is l p , p ∈ [1, ∞), we establish a necessary and sufficient condition for the matrix operator in l p defined by the matrix A to have a bounded inverse on l p .

Structure properties of W matrices

We continue an investigation of the class W of real square matrices. A matrix belongs to W if and only if certain pairs of its complementary cones intersect in the zero vector only. We show that a W matrix with no zero diagonal entries is a P 0 matrix (each principal minor is nonnegative), and a W matrix with no nonzero diagonal entries is a nonnegative matrix which either is monomial or has a zero column. Lastly, we show how these two results impose certain necessary conditions on the structure of a W matrix.

A Remark on Perfect Gaussian Elimination of Symmetric Matrices

Let M be a symmetric matrix with non-zero diagonal entries. A result of Golumbic [3, 5] states that if M has a perfect elimination scheme then it also has a perfect elimination scheme with the additional property that all pivots are chosen along the main diagonal. However, the proof given by Golumbic seems to be incomplete. In the present note, we refine Golumbic's proof, thus obtaining a complete version of it.