On The Solution Space Of A Class Of Matrix Equations

There have been many in-depth studies on the theory and application of solution space of matrix equation at China and abroad,and M.Asunci′on Beita considered the dimensionality of solution space of matrix equation(AX-XA)W = 0 in 1985,that is,the dimensionality of solution space of matrix equation AX-XA = 0 restricted to the subspace W.Through some restrictions,the dimensional range of the solution space of the matrix equation(AX-XA)W = 0is obtained when A is a diagonal matrix.In addition,Minhao Zhao extended the diagonal matrix condition to a more general Jordan-type matrix by vectorization method in 2021,and obtained the dimensional formula of the solution space of the matrix equation(AX-XA)W = 0 when W is a one-dimensional subspace and when W is a subspace spanned by some n-dimensional unit vectors.In this paper,the matrix equation(AX-XA)W = 0 is reduced to the dimensionality of the solution space for solving the matrix equation R(JX-XJ)= 0,where R is the reduced row echelon matrix,and J is Jordan-type matrix.Through the typical linear algebraic method,especially the vectorization method for matrix equation,the matrix equation is transformed into a general linear equation system,and taking full advantage of the characteristics of the image of a class of important linear transformations,as well as the skills and methods of matrix partitioning,we obtain the dimensional expression of solution space of matrix equation R(JX-XJ)= 0.Thus the existing relevant conclusions are generalized and the relevant research results are enriched.