Further Solutions of a Yang-Baxter-like Matrix Equation

2013 ◽  
Vol 3 (4) ◽  
pp. 352-362 ◽  
Author(s):  
Jiu Ding ◽  
Chenhua Zhang ◽  
Noah H. Rhee
Keyword(s):  
Canonical Form ◽  
Infinite Number ◽  
Matrix Equation ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Square Matrix

AbstractThe Yang-Baxter-like matrix equation AXA = XAX is reconsidered, and an infinite number of solutions that commute with any given complex square matrix A are found. Our results here are based on the fact that the matrix A can be replaced with its Jordan canonical form. We also discuss the explicit structure of the solutions obtained.

scholarly journals A New Solution to the Matrix EquationX−AX¯B=C

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Caiqin Song
Keyword(s):  
Unique Solution ◽  
Canonical Form ◽  
Matrix Equation ◽  
Explicit Solution ◽  
Symmetric Operator ◽  
Operator Matrix ◽  
Numerical Example ◽  
Controllability Matrix ◽  
The Matrix

We investigate the matrix equationX−AX¯B=C. For convenience, the matrix equationX−AX¯B=Cis named as Kalman-Yakubovich-conjugate matrix equation. The explicit solution is constructed when the above matrix equation has unique solution. And this solution is stated as a polynomial of coefficient matrices of the matrix equation. Moreover, the explicit solution is also expressed by the symmetric operator matrix, controllability matrix, and observability matrix. The proposed approach does not require the coefficient matrices to be in arbitrary canonical form. At the end of this paper, the numerical example is shown to illustrate the effectiveness of the proposed method.

The Structure and Distribution of Roots of 2×2 Matrices

2013 ◽  
Vol 765-767 ◽  
pp. 667-669
Author(s):  
Yuan Yuan Li
Keyword(s):  
Infinitely Many Solutions ◽  
Sufficient Condition ◽  
Singular Matrix ◽  
Necessary And Sufficient Condition ◽  
Number Of Solutions ◽  
Jordan Canonical Form ◽  
The Matrix ◽  
Explicit Formulae ◽  
Necessary And Sufficient ◽  
Distribution Of Roots

This paper is concerned with Jordan canonical form theorem of algebraic formulae giving all the solutions of the matrix equation Xm= A where n is a positive integer greater than 2 and A is a 2 × 2 matrix with real or complex elements. If A is a 2 × 2 non-singular matrix, the equation Xm = A has infinitely many solutions and we obtain explicit formulae giving all the solutions. If A is a 2 × 2 singular matrix, and we obtained necessary and sufficient condition of square root . This leads to very simple formulae for all the solutions when A is either a singular matrix or a non-singular matrix with two coincident eigenvalues. We also determine the precise number of solutions in various cases.

On the Solutions of the Matrix Equation f(X,X*)=g(X,X*)

1972 ◽  
Vol 15 (1) ◽  
pp. 45-49
Author(s):  
P. Basavappa
Keyword(s):  
Matrix Equation ◽  
Identity Matrix ◽  
Normal Matrices ◽  
Special Cases ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Matrix Identities ◽  
Square Matrix

It is well known that the matrix identities XX*=I, X=X* and XX* = X*X, where X is a square matrix with complex elements, X* is the conjugate transpose of X and I is the identity matrix, characterize unitary, hermitian and normal matrices respectively. These identities are special cases of more general equations of the form (a)f(X, X*)=A and (b)f(Z, X*)=g(X, X*) where f(x, y) and g(x, y) are monomials of one of the following four forms: xyxy…xyxy, xyxy…xyx, yxyx…yxyx, and yxyx…yxy. In this paper all equations of the form (a) and (b) are solved completely. It may be noted a particular case of f(X, X*)=A, viz. XX'=A, where X is a real square matrix and X' is the transpose of X was solved by WeitzenbÖck [3]. The distinct equations given by (a) and (b) are enumerated and solved.

An application of the Jordan canonical form to the epidemic problem

1980 ◽  
Vol 17 (02) ◽  
pp. 313-323
Author(s):  
Linda P. Gilbert ◽  
A. M. Johnson
Keyword(s):  
Probability Distribution ◽  
Canonical Form ◽  
Difference Equations ◽  
Jordan Canonical Form ◽  
Stochastic Version ◽  
The Matrix

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

An application of the Jordan canonical form to the epidemic problem

1980 ◽  
Vol 17 (2) ◽  
pp. 313-323 ◽  
Author(s):  
Linda P. Gilbert ◽  
A. M. Johnson
Keyword(s):  
Probability Distribution ◽  
Canonical Form ◽  
Difference Equations ◽  
Jordan Canonical Form ◽  
Stochastic Version ◽  
The Matrix

The stochastic version of the simple epidemic is considered in this paper. A method which has computational advantages is developed for computing the probability distribution of the process. The method is based on the Jordan canonical form of the matrix representing the system of differential-difference equations for the simple epidemic. It is also shown that the method can be applied to the general epidemic as well as any other right-shift process.

Rank r Solutions to the Matrix Equation XAXT = C, A Nonalternate, C Alternate, Over GF(2y).

1974 ◽  
Vol 26 (1) ◽  
pp. 78-90 ◽  
Author(s):  
Philip G. Buckhiester
Keyword(s):  
Finite Field ◽  
Matrix Equation ◽  
Symmetric Matrices ◽  
Number Of Solutions ◽  
The Matrix ◽  
Image Position

Let GF(q) denote a finite field of order q = py, p a prime. Let A and C be symmetric matrices of order n, rank m and order s, rank k, respectively, over GF(q). Carlitz [6] has determined the number N(A, C, n, s) of solutions X over GF(q), for p an odd prime, to the matrix equation1.1where n = m. Furthermore, Hodges [9] has determined the number N(A, C, n, s, r) of s × n matrices X of rank r over GF(q), p an odd prime, which satisfy (1.1). Perkin [10] has enumerated the s × n matrices of given rank r over GF(q), q = 2y, such that XXT = 0. Finally, the author [3] has determined the number of solutions to (1.1) in case C = 0, where q = 2y.

scholarly journals On the Canonical Form of a Rational Integral Function of a Matrix

1932 ◽  
Vol 3 (2) ◽  
pp. 135-143 ◽  
Author(s):  
D. E. Rutherford
Keyword(s):  
Canonical Form ◽  
Integral Function ◽  
The Other ◽  
Singular Matrix ◽  
The Matrix ◽  
Image Position ◽  
Rational Integral ◽  
Square Matrix

It is well known that the square matrix, of rank n−k + 1,which we shall denote by B where any element to the left of, or below the nonzero diagonal b1, k, b2, k + 1, . …, bn−k + 1, n is zero, can be resolved into factors Z−1DZ; where D is a square matrix of order n having the elements d1, k, d2, k + 1, . …, dn−k + 1, n all unity and all the other elements zero, and where Z is a non-singular matrix. In this paper we shall show in a particular case that this is so, and in the case in question we shall exhibit the matrix Z explicitly. Application of this is made to find the classical canonical form of a rational integral function of a square matrix A.

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