The reflexive solutions of the matrix equation AX B = C

AbstractIn this paper, we study the existence of a reflexive, with respect to the generalized reflection matrix P, solution of the matrix equation AX B = C. For the special case when B = I, we get the result of Peng and Hu [1]

Block Decompositions and Applications of Generalized Reflexive Matrices

Generalize reflexive matrices are a special class of matrices that have the relation where and are some generalized reflection matrices. The nontrivial cases ( or ) of this class of matrices occur very often in many scientific and engineering applications. They are also a generalization of centrosymmetric matrices and reflexive matrices. The main purpose of this paper is to present block decomposition schemes for generalized reflexive matrices of various types and to obtain their decomposed explicit block-diagonal structures. The decompositions make use of unitary equivalence transformations and, therefore, preserve the singular values of the matrices. They lead to more efficient sequential computations and at the same time induce large-grain parallelism as a by-product, making themselves computationally attractive for large-scale applications. A numerical example is employed to show the usefulness of the developed explicit decompositions for decoupling linear least-square problems whose coefficient matrices are of this class into smaller and independent subproblems

The Least Squares Hermitian (Anti)reflexive Solution with the Least Norm to Matrix Equation A

For a given generalized reflection matrix J, that is, JH=J, J2=I, where JH is the conjugate transpose matrix of J, a matrix A∈Cn×n is called a Hermitian (anti)reflexive matrix with respect to J if AH=A and A=±JAJ. By using the Kronecker product, we derive the explicit expression of least squares Hermitian (anti)reflexive solution with the least norm to matrix equation AXB=C over complex field

Fast Algorithms for Solving FLS R

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS R-factor block circulant (retrocirculant) matrix over field F. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field F is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of AX=b in the class of the level-2 FLS (R,r)-circulant(retrocirculant) matrix of type (m,n) over field F are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms

Graphs and networks theory

This chapter discusses graphs and networks theory

Causal spaces and the application of critical point theory to general relativity

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