Certain Types Of Matrices Research Articles

The smallest singular value anomaly: The reasons behind sharp anomaly

Abstract Let A A be an arbitrary matrix in which the number of rows, m m , is considerably larger than the number of columns, n n . Let the submatrix A i , i = 1 , … , m {A}_{i},\hspace{0.33em}i=1,\ldots ,m , be composed from the first i i rows of A A , and let β i {\beta }_{i} denote the smallest singular value of A i {A}_{i} . Recently, we observed that the first part of this sequence, β 1 , … , β n {\beta }_{1},\ldots ,{\beta }_{n} , is descending, while the second part, β n , … , β m {\beta }_{n},\ldots ,{\beta }_{m} , is ascending. This property is called “the smallest singular value anomaly.” In this article, we expose another interesting feature of this sequence. It is shown that certain types of matrices possess the sharp anomaly phenomenon: First, when i i is considerably smaller than n n , the value of β i {\beta }_{i} decreases rather slowly. Then, as i i approaches n n from below, there is fast reduction in the value of β i {\beta }_{i} , making β n {\beta }_{n} much smaller than β 1 {\beta }_{1} . Yet, once i i passes n n , the situation is reversed and β i {\beta }_{i} increases rapidly. Finally, when i i moves away from n n , the rate of increase slows down. The article illustrates this behavior and explores its reasons. It is shown that the sharp anomaly phenomenon occurs in matrices with “scattering rows.”

FINDING THE SOLUTION OF THE ELLIPTIC SYSTEM OF THE FIRSTORDER IN THE FORM OF A VECTOR-FUNCTION

The relationship between Douglis analytic functions and the analytic functions of the Laplace equation for elliptic equations with constant coefficients serves the same purpose. In this sense, some boundary value problems solved by Douglis using analytic functions remain relevant today. Douglis’s Bitsadze-Samarsky problem on analytic functions was reconsidered by A.P. Soldatov. Meanwhile, for certain types of matrices and regions, V.G. Nikolaev proved the existence and uniqueness of the Schwarz problem solution. In any region G of the complex plane C, we considered a first order system y J x =F, where the eigenvalues of the constant matrix J Cl l lie in the upper half-plane, Imv > 0. In the case where l = 1 J = J(z) is continuous and ImJ > 0, we obtain the Beltrami equation. In this paper, to find the solution of elliptic equation with constant coefficient, the corresponding J-analytic system was constructed by choosing the matrix J of size 2 x 2andJ has no real eigenvalues. In the postcomponential notation, the system consists of two differential equations depending on the variable z. Thus, the solution of this equation was found using the vector- function.

Finding Saddle-Node Bifurcations via a Nonlinear Generalized Collatz–Wielandt Formula

The Collatz–Wielandt formula obtained by Lothar Collatz (1942) and Helmut Wielandt (1950) provides a simple variational characterization of the Perron–Frobenius eigenvalue of certain types of matrices. We introduce a nonlinear generalization of the Collatz–Wielandt formula and show that it enables finding the so-called maximal saddle-node bifurcations of systems of nonlinear equations. As a consequence, we obtain an easily verifiable criterion for the detection of such saddle-node bifurcations. We illustrate our approach by examples of finite-difference approximations of nonlinear partial differential equations and a system of power flow.

On the solution of the Schwarz problem for J-analytic functions in a domain bounded by a Lyapunov contour

We consider a second-order elliptic system of partial differential equations in a special case. The results can be used to study the solvability of the Schwarz problem for J-analytic functions for certain types of matrices J.

Counting of Binary Matrices Avoiding Some 2 × 2 Matrices

The number of matrices avoiding certain types of matrices is NP-hard in general. In this paper the binary matrices are considered. In particular, the problem of finding the total number of special binary matrices avoiding some types of 2 × 2 matrices is the main objective of this paper. The solution of the problem is given under some constraints as well as under general situation. The formula for the special binary matrices is obtained for total count of matrices of order n × k and also obtained the formula for special binary matrices avoiding some matrices of order 2 × 2. The formula is obtained in terms of the Catalan numbers.

Complements to some results on eigenvalues and singular values of matrices

In this article we study eigenvalues and singular values of matrices. We provide some majorization inequalities for certain types of matrices. For this end we use group-induced cone orderings and normal maps.

The Perron-Frobenius theorem: some of its applications

The Perron-Frobenius theorem provides a simple characterization of the eigenvectors and eigenvalues of certain types of matrices with nonnegative entries. The importance of the Perron-Frobenius theorem stems from the fact that eigenvalue problems on these types of matrices frequently arise in many different fields of science and engineering. In this article, the authors discuss the applications of this theorem in diverse areas such as steady state behavior of Markov chains, power control in wireless networks, commodity pricing models in economics, population growth models, and Web search engines. The article starts with a review and discussion of the mathematical foundations.

An application of graph theory to group theory

An application of graph theory to group theory is presented. A problem of finding the eigenvectors for certain types of matrices is considered.

On the ranks of certain matrices

A few basic results regarding the ranks of certain types of matrices are presented. Also, some bounds on the ranks of the real and imaginary parts of a positive semi-definite Hermitian matrix are worked out. Applications of particular cases of these results in the areas like passive network synthesis, generation of two-variable reactance matrices, etc., are pointed out.

Explicit quasispectral decompositions of certain types of matrices

L e t ' A ---[A~.i ~,)=1 = [[a~;o~]o,~=l]~j=l b e a b l o c k m a t r i x o f o r d e r rm w i t h b l o c k s of o r d e r m and V ~ m n s u p p o s e t h a t w i t h t h e a i d o f t h e m a t r i x U = [ kj]~,j=~ = [ [ ~ . ~ i o z ] z , ~ = x ] ~ , i = l i t c a n b e r e d u c e d t o q u a s i / A ~,~ ,~ n d i a g o n a l f o r m * A = ( in '=l = {[%jo~]o,~=~}i=~ , i . e . , o n e h a s A = U A U -~. (1 ) Equation (i) will be called a quasispectral decomposition of the matrix A if explicit expressions are known for the blocks Aj and Ukj. In this paper, explicit quasispectral decompositions are given for certain types of matrices met with in problems of mechanics. They can be used also as test problems for the development of computational methods of linear algebra for matrices of high order. Here too sufficient conditions for the existence and the form of explicit quasispectral and spectral decompositions of block matrices with blocks of one order are given. i. Quasicyclic Matrices i.i. Nonsymmetric Quasicyclic Matrices. A square quasimodular% matrix will be called quasicyclic if the blocks of its p-th quasisuperdiagonal coincide with the blocks of the (n -p)-th quasisubdiagonal (p = 1 .... , n -i) ;

A computational algorithm for inners

A Gaussian elimination algorithm is proposed to compute the determinants of inners of certain types of matrices. This algorithm applies to the clustering of roots as well as to the roots distribution of a given polynomial. Several examples on stability, relative stability, and aperiodicity for both continuous and discrete systems are presented.

XVI.—Some Continuant Determinants arising in Physics and Chemistry—II

SynopsisSome of the formulae obtained in this paper are likely to find application in problems concerning a rectangular lattice of “atoms”, each of which is under the influence of its near neighbours. Some of the determinants considered apply to cases in which both the nearest and the next nearest neighbours are operative. The inverses of certain types of matrices are found, and these may prove to be of value either in solving systems of linear equations such as arise in relaxation problems, or in determining the latent roots of matrices which may occur in problems in applied mathematics.

A rapid method of inversion of certain types of matrices

A rapid method of inversion of certain types of matrices