Linear transformations that preserve the assignment (original) (raw)
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On some new matrix transformations
Journal of Inequalities and Applications, 2013
In this paper, we characterize some matrix classes (ω(p, s), V λ σ ), (ω p (s), V λ σ ) and (ω p (s), V λ σ ) reg under appropriate conditions.
On The Linear Transformation of Division Matrices
In this study, we deal with functions from the square matrices to square matrices, which the same order. Such a function will be called a linear transformation, defined as follows: Let M n (R) be a set of square matrices of order n, n ϵ S, and A be regular matrix in M n (R), then the special function T A : M n (R) → M n (R) A X X T X A is called a linear transformation of M n (R) to M n (R) the following two properties are true for all X,Y ϵ M n (R), and scalars α ϵ R: i. T A (X+Y) = T A (X) + T A (Y). (We say that T A preserves additivity) ii. T A (αX)= αT A (X) (We say that T A preserves scalar multiplication) In this case the matrix A is called the standard matrix of the function T A. Here, we transfer some well known properties of linear transformations to the above defined elements in the set all { T A : A regular in M n (R)} [1].
Special Issue on “Structured Matrices: Analysis, Algorithms and Applications”
Linear Algebra and its Applications
The mathematical modeling of problems of the real world often leads to problems in linear algebra involving structured matrices where the entries are defined by few parameters according to a compact formula. Matrix patterns and structural properties provide a uniform means for describing different features of the problem that they model. The analysis of theoretical and computational properties of these structures is a fundamental step in the design of efficient solution algorithms. Certain structures are encountered very frequently and reflect specific features that are common to different problems arising in diverse fields of theoretical and applied mathematics and engineering. In particular, properties of shift invariance, shared by many mathematical entities like point-spread functions, integral kernels, probability distributions, convolutions, etc., are the common feature which originates Toeplitz matrices. In fact, Toeplitz matrices, characterized by having constant entries along their diagonals, are encountered in fields like image processing, signal processing, digital filtering, queueing theory, computer algebra, linear prediction and in the numerical solution of certain difference and differential equations, just to mention a few. The interest in this class of matrices is not motivated only by the applications; in fact, Toeplitz matrices are endowed with a very rich set of mathematical properties and there exists a very wide literature dated back to the first half of the last century on their analytic, algebraic, spectral and computational properties. Other classes of structured matrices are less pervasive in terms of applications but nevertheless they are not less important. Frobenius matrices, Hankel matrices, Sylvester matrices and Bezoutians, encountered in control theory, in stability issues, and in polynomial computations have a rich variety of theoretical properties and have been object of many studies. Vandermonde matrices, Cauchy matrices, Loewner matrices and Pick matrices are more frequently encountered in the framework of interpolation problems. Tridiagonal and more general banded matrices and their inverses, which are semiseparable matrices, are very familiar in numerical analysis. Their extension to more general classes and the design of efficient algorithms for them has recently received much attention. Multi-dimensional problems lead to matrices which can be represented as structured block matrices with a structure within the blocks themselves. Kronecker product
Matrices of zeros and ones with fixed row and column sum vectors
Linear Algebra and Its Applications, 1980
Let m and n be positive integers, and let R = (rl,. . . , r,) and S = (an.. . ,sn) be nonnegative integral vectors. We survey the combinatorial properties of the set of all m x n matrices of O's and l's having ri l's in row i and si I's in column i. A number of new results are proved. The results can also be formulated in terms of the set of bipartite graphs with a bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S. of Fu(R,S) . 1 d' g mc u m some recent results. In doing so, we present in some cases new proofs of theorems which may be more transparent than those in the literature. Also there appear here for the first time a number of new results, notably the solution (Theorem 6.8) of a problem posed by Ryser [56, p. 761 in 1963. Other new results include Theorems 3.10, 4.2, 4.4, 5.8, 5.9, 6.8, 6.10, 7.3, 8.3, and 8.13, and Corollaries 5.6 and 8.6. Over twenty problems are proposed. Before proceeding we give two alternative interpretations of 8(R, S). Let X=(x,,..., xm} and Y={ yr,..., y,} be disjoint sets of m and n elements, respectively. Let BG(R,S) d enote the collection of all bipartite graphs G with the following properties: (BGl) The vertices of G are xi,. . . , x,, yl,. . . , y,,. (BG2) Each edge of G joins a vertex in X to a vertex in Y. (BG3) The degree (or valency) of xi is r, for i = 1,. . . , m, and the degree of yj is si for j=l,...,n. Then there is a one-to-one correspondence between the matrices in '%(R,S) and the bipartite graphs in BG(R, S), determined as follows. If A =[a,] E
Advances in Linear Algebra Matrix Theory, 2012
This paper considers rank of a rhotrix and characterizes its properties, as an extension of ideas to the rhotrix theory rhomboidal arrays, introduced in 2003 as a new paradigm of matrix theory of rectangular arrays. Furthermore, we present the necessary and sufficient condition under which a linear map can be represented over rhotrix.
Applications of Linear Transformations to Matrix Equations
Linear Algebra and Its Applications, 1997
We consider the linear transformation T(X) = AX -CXB where A, C E M,, B E M,. We show a new approach to obtaining conditions for the existence and uniqueness of the solution X of the matrix equation T(X) = R. As a consequence of our approach we present a simple characterization of a full-rank solution to the matrix equation. We apply the existence theorem to a general form of the observer matrix equation and characterize the existence of a full-rank solution. 0 1997 Elsevier Science Inc. NOTATION AND KEY WORDS The following symbols and key words are used in this paper: M cl0 Set of n-by-m complex matrices; M,,, = M,. Column space of X E M,, k. N(T) Null space of a linear transformation T : M,, s -+ M,, $. A o B Kronecker product of matrices A and B [lo, Chapter 41. A @ B Direct sum of matrices A and B [9, Chapter 01. LINEARALGEBBAANDITSAPPLICATIONS 267:221-240(1997) 0 1997 Elsevier Science Inc. All rights reserved. ,, where ej E @" is in the ith column, i.e., E, I = [e, 0 *em O] E M,.
Linear Algebra and its Applications 363 (2003) 65--80
We present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions. Our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices, but we also obtain new results for nonstrict inequalities of the spectral radii. We emphasize two different approaches, one combinatorial and the other analytic and discuss their merits in the light of the results obtained. We try to get fairly general results and indicate by counter-examples that some of our hypotheses cannot be relaxed in certain directions.
A Bruhat order for the class of (0,1)-matrices with row sum vector R and column sum vector S
Electronic Journal of Linear Algebra, 2005
Generalizing the Bruhat order for permutations (so for permutation matrices), a Bruhat order is defined for the class of m by n (0, 1)-matrices with a given row and column sum vector. An algorithm is given for constructing a minimal matrix (with respect to the Bruhat order) in such a class. This algorithm simplifies in the case that the row and column sums are all equal to a constant k. When k = 2 or k = 3, all minimal matrices are determined. Examples are presented that suggest such a determination might be very difficult for k ≥ 4.
Mat 320-Linear Algebra I Summary of Lectures
2012
Linear equations, translations to matrices, pages 4-6 in [1], [2] p'2. 1.2 January 11 Elementary row operations on matrices, Row Echelon Form (REF) of a matrix, solutions of system of linear equations using REF.Equivalent systems of equations. Lead and free variables. Pages 7-12 in [1], [2] pages 3-9, without matrix notation. 1.3 January 13 Reduced row echelon form of a matrix. How to find solutions of systems of equations using reduced row echelon form. Pages 13-14 in [1], Pages 46-49 [2]. Discussed system of homogeneous equations. Showed that for m homogeneous equations in n unknowns, with m < n one has always a nontrivial solutio