Equation AX Research Articles

An explicit solution to the matrix equation AX − XF = BY

A complete, general and explicit solution to the generalized Sylvester matrix equation AX − XF = BY, with the matrix F in a companion form, is proposed. The solution is in an extremely neat form represented by a symmetric operator matrix, a Hankel matrix and the controllability matrix of the matrix pair ( A, B). Furthermore, several equivalent forms of this solution are also presented. Based on these presented results, explicit solutions to the normal Sylvester equation and the well-known Lyapunov matrix equation are also established. The results provide great convenience to the analysis of the solution to the equation, and can perform important functions in many analysis and design problems in control systems theory. As a demonstration, a simple and effective approach for parametric pole assignment is proposed.

The Klein Quartic as a Cyclic Group Generator

Let k be a field, and let a, b, c be three elements in k ◊ . The nonsingular projective plane curve X defined over k with equation ax 3 y + by 3 z + cz 3 x = 0 has genus 3 and is reduced to the familiar Klein quartic when a = b = c = 1. The Jacobian JX of X is a three-dimensional abelian variety, defined over k as well. The aim of this article is to give some formulas for the number of points of the group JX(k) of rational points of JX if k = Fq is a finite field. We assume that the full group of seventh roots of unity is contained in k; this amounts to saying that q 1 (mod 7). If q is a prime number and the coecients a, b, c are appropriately chosen, we noticed that the number of points of the group JX(k) is prime in a significant number of occurences. This provides cyclic groups which seem to be accurate for cryptographic applications. 2000 Math. Subj. Class. 11G25.

On the Hermitian-Generalized Hamiltonian Solutions of Linear Matrix Equations

By means of the properties of Hermitian-generalized Hamiltonian matrices, we give some necessary and sufficient conditions for the solvability of the matrix equation AX = B in Hermitian-generalized Hamiltonian matrix set $HHC^{n\times n}$. In the case where AX = B is solvable in $HHC^{n\times n}$, we derive the general representation of the solutions. We also give the expression of the solution for the corresponding optimal approximation problem.

Solution of linear matrix equations in a *-congruence class

The possible *congruence classes of a square solution to the real or complex linearmatrix equation AX = B are determined. The solution is elementary and self contained, and includesseveral known results as special cases, e.g., X is Hermitian or positive semidefinite, and X is realwith positive definite symmetric part.

Existence of a low rank or ℋ︁‐matrix approximant to the solution of a Sylvester equation

AbstractWe consider the Sylvester equation AX−XB+C=0 where the matrix C∈ℂn×m is of low rank and the spectra of A∈ℂn×n and B∈ℂm×m are separated by a line. We prove that the singular values of the solution X decay exponentially, that means for any ε∈(0,1) there exists a matrix X̃ of rank k=O(log(1/ε)) such that ∥X−X̃∥2⩽ε∥X∥2. As a generalization we prove that if A,B,C are hierarchical matrices then the solution X can be approximated by the hierarchical matrix format described in Hackbusch (Computing 2000; 62: 89–108). The blockwise rank of the approximation is again proportional to log(1/ε). Copyright © 2004 John Wiley & Sons, Ltd.

Solving a kind of restricted matrix equations and Cramer rule

The problem of finding determinantal formula for solutions of some restricted linear systems Ax= b has been discussed in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 116 (1989) 27, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303]. This paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equations AX=D (R(X)⊂R(A k 1 )), XB=D (N(X)⊃N(B k 2 )), AXB=D (R(X)⊂R(A k 1 ),N(X)⊃N(B k 2 )), where A∈ C n× n with Ind( A)= k 1, B∈ C m× m with Ind( B)= k 2, and D∈ C n× m . The results in [Linear Multilinear Algebra 15 (1984) 319, Linear Algebra Appl. 74 (1986) 213, Linear Multilinear Algebra 34 (1993) 177, Appl. Math. Comput. 125 (2002) 303] are partially the special cases in our paper. The classic Cramer rule is also a special case of our results.

A note on commutativity up to a factor of bounded operators

In this note, we explore commutativity up to a factor AB = λBA for bounded operators A and B in a complex Hilbert space. Conditions on possible values of the factor A are formulated and shown to depend on spectral properties of the operators. Commutativity up to a unitary factor is considered. In some cases, we obtain some properties of the solution space of the operator equation AX = λxA and explore the structures of A and B that satisfy AB = λBA for some λ ∈ C {0}. A quantum effect is an operator A on a complex Hilbert space that satisfies 0 ≤ A ≤ I. The sequential product of quantum effects A and B is defined by A o B = A 1 /2 BA 1 /2. We also obtain properties of the sequential product.

The reflexive and anti-reflexive solutions of the matrix equation AX= B

An n× n complex matrix P is said to be a generalized reflection matrix if P H = P and P 2= I. An n× n complex matrix A is said to be a reflexive (or anti-reflexive) matrix with respect to the generalized reflection matrix P if A= PAP (or A=− PAP). This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the reflexive and anti-reflexive with respect to a generalized reflection matrix P solutions of the matrix equation AX= B. In addition, in corresponding solution set of the equation, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm have been provided.

The Diophantine equation ax2 + 2bxy − 4ay2 = ±1

We discuss, with the aid of arithmetical properties of the ring of the Gaussian integers, the solvability of the Diophantine equation ax2 + 2bxy − 4ay2 = ±1, where a and b are nonnegative integers. The discussion is relative to the solution of Pell′s equation v2 − (4a2 + b2)w2 = −4.

Computing tall skinny solutions of AX− XB= C

We will concentrate on the numerical computation of so-called tall and skinny solutions X of the Sylvester equations AX− XB= C. By this we mean that A is an n× n matrix with cheaply applicable action (A is for example sparse), and B a k× k matrix, with k≪ n. This type of Sylvester equation plays an important role in the computation of invariant subspaces when block Rayleigh quotient or block Jacobi–Davidson methods are used. There exist essentially two types of subspace projection methods for such equations. One in which the tall skinny matrices are treated as rigid objects, and one in which they are treated as a basis of a subspace. In this paper we review the two different types, design new methods, and aim to identify which application should make use of which solution method.

A conjecture concerning the exponential diophantine equation ax+by=cz

A conjecture concerning the exponential diophantine equation a^x+b^y=c^z

Estimation of symmetric positive-definite matrices from imperfect measurements

In a number of contexts relevant to control problems, including estimation of robot dynamics, covariance, and smart structure mass and stiffness matrices, we need to solve an overdetermined set of linear equations AX /spl ap/ 8 with the constraint that the matrix X be symmetric and positive definite. In the classical least-squares method, the measurements of A are assumed to be free of error, hence, all errors are confined to B. Thus, the optimal solution is given by minimizing the optimization criterion /spl par/AX /spl par//sub F//sup 2/. However, this assumption is often impractical. Sampling errors, modeling errors, and, sometimes, human errors bring inaccuracies to A as well. In this note, we introduce a different optimization criterion, based on area, which takes the errors in both A and B into consideration. Under the condition that the data matrices A and B are full rank, which in practice is easy to satisfy, the analytic expression of the global optimizer is derived. A method to handle the case that A is full rank and B loses rank is also discussed. Experimental results indicate that the new approach is practical, and improves performance.

On Le's and Bugeaud's Papers about the Equation ax 2 + b 2m−1 = 4 c p

We correct an inaccuracy in the papers of Le, Mignotte, and Bugeaud on the generalized Ramanujan-Nagell equation.

Geometrical solution of some cubic equations with non-real roots

A general cubic equation ax 3 + bx 2 + cx + d = 0 where a, b, c, d ∈R, a ≠ 0 has three roots with two possibilities—either all three roots are real or one root is real and the remaining two roots are imaginary. Dealing with the second possibility this paper attempts to give the geometrical locations of the imaginary roots of the equation under three different sets of conditions. These sets of conditions include: (i) the real root of the given cubic equation is given, (ii) the real part of an imaginary root is given, and (iii) the imaginary part of an imaginary root is given.

UNITARY INTERPOLATION FOR OPERATORS IN TRIDIAGONAL ALGEBRAS

Given operators X and Y acting on a Hilbert space H, an interpolating operator is a bounded operator A such that AX = Y. An interpolating operator for the n-operators satisfies the equation AX<TEX>$\_$</TEX>i/ : Y<TEX>$\_$</TEX>i/, for i = 1, 2 …, n. In this article, we obtained the following : Let X = (x<TEX>$\_$</TEX>ij/) and Y = (y<TEX>$\_$</TEX>ij/) be operators acting on H such that <TEX>$\varkappa$</TEX><TEX>$\_$</TEX> i<TEX>$\sigma$</TEX> (i)/ 0 for all i. Then the following statements are equivalent. (1) There exists a unitary operator A in Alg(equation omitted) such that AX = Y and every E in (equation omitted) reduces A. (2) sup{(equation omitted)}<<TEX>$\infty$</TEX> and (equation omitted) = 1 for all i = 1, 2, ….

Interpolation in self-adjoint settings

We study the operator equation AX = Y, where the operators X and Y are given and the operator A is required to lie in some von Neumann algebra. We derive a necessary and sufficient condition for the existence of a solution A. The condition is that there must exist a constant K so that, for all finite collections of operators {D 1 , D 2 ,..., D n } in the commutant, and all collections of vectors {f 1 , f 2 ,..., f n }, we have ∥Σ n j=1 D j Yf j ∥ ≤ K ∥ Σ n j=1 D j X f j ∥. We also study the equality ∥DY f ∥ = K∥DX f ∥, in connection with solving the equation AX = Y where the operator A is required to lie in some CSL algebra.

Block Krylov subspace methods for solving large Sylvester equations

In the present paper, we propose block Krylov subspace methods for solving the Sylvester matrix equation AX−XB=C. We first consider the case when A is large and B is of small size. We use block Krylov subspace methods such as the block Arnoldi and the block Lanczos algorithms to compute approximations to the solution of the Sylvester matrix equation. When both matrices are large and the right-hand side matrix is of small rank, we will show how to extract low-rank approximations. We give some theoretical results such as perturbation results and bounds of the norm of the error. Numerical experiments will also be given to show the effectiveness of these block methods.

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.

Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax 2 + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.

A Diophantine equation appearing in Diophantine approximation

All Diophantine equations ax 2 + by 2 + cz 2 = 1 + dxyz, with a, b, c, d ∈ N and a| d, b| d, c| d, having solutions ( x, y, z) ∈ N 3 are determined. One particular equation of this type 2 x 2 + 2 y 2 + 3 z 2 = 1 + 6 xyz appeared recently in connection with the description of the lower part of the approximation spectrum for quaternions.