Dimension Formulas Of The Solution Space Of A Class Of Matrix Equations

In modern mathematics, matrix obviously has been an useful tool in the theory research and practical application, and usually read in the mathematic subjects of advanced algebra and statistics. However, in the research of matrix, matrix equation plays a very important role, and extensively applies to algebra, combinatorial, graph theory, control and other fields.In recent years, matrix equation this matrix equation. Then we get some results on the solution of the matrix equation by using these formulas.In modern mathematics, matrix obviously has been an useful tool in the theory research and practical application, and usually read in the mathematic subjects of advanced algebra and statistics. However, in the research of matrix, matrix equation plays a very important role, and extensively applies to algebra, combinatorial, graph theory, control and other fields.In recent years, matrix equation EDCXAXBT=+ has been getting more and more attention as its significant theoretical value and promising application prospect, mathematics scientists have already done relevant research on the existence of solution and algorithm of this equation, but as an essential issue in the research of matrix equation, the problem of the dimension of solution space of homogeneous form of this equation was researched little.Through verifying this problem in large number of aspects, it’s clear that this problem is too hard to deal with at once, additionally, particular case could reflect the characteristic of thinking methods, then we turn around to research the particular situation of this equation, namely, the problem of the dimension of solution space of the following matrix equation, =-0TTTABXAXB ω In which ω=1 and ω-=1.By using Kronecker canonical form of matrix pencil( A, B), this paper mainly focuses on the issue of the dimension of the solution space of the matrix equation =-0TTTABXAXB ω in which,m nA B×∈￡ and ω =1 and ω=-1, then get the formulas of the dimension of the solution space of this kind of matrix equation.Firstly, by using Kronecker canonical form of matrix pencil( A, B), we could decompose the matrix equation into two paratactic systems of linear equations T T T0 i ii i i ii iA X B-ωB X A = T T T T T T0()0i ij j i ji j j ji i j ij i A X B B X Ai jA X B B X Aωω?- =??≠- =?? Based on dimensional formulas of solution spaces of the two systems of linear equations and the relation of the solution spaces between the matrix equation and the two systems, we will obtain the dimensional formulas of the solution space of this matrix equation. Then we get some results on the solution of the matrix equation by using these formulas.