The Solution Space Of A Class Of Matrix Equations

The theory and applications of matrix equations have been studied in depth and systemat-ically in China and abroad.Matrix equations have been widely used in many scientific fields.In 1985 M.Asuncin Beitia considered the problem of the dimensionality of the solution space of the matrix equation(AX-XA)W=0,i.e.,the dimensionality of the solution space of the matrix equation AX-XA=0 restricted to the sub space W.The dimension of the solution space of the matrix equation(AX-XA)W=0 when A is a diagonal matrix is obtained by computing the matrix subspace centralized with the diagonal matrix.In this paper,starting from the results of Frobenius’ study on the dimensionality of the centralized subspace of a given matrix,the matrix equation(AX-XA)W=0 is transformed into a linear system of equations for Jordan type matrix A by vectorization method.Thus,the dimension of the solution space of the matrix equation(AX-XA)W=0 can be obtained by calculating the rank of its coefficient matrix.We obtain the dimension of the solution space of the matrix equation(AX-XA)W=0 when W is a one-dimensional subspace.And we obtain the formula for the dimension of the solution space of the matrix equation(AX-XA)W=0 when W is a subspace spanned by some n-dimensional unit vectors.Thus,the existing diagonal matrix condition is extended to more general Jordan-type matrices,and the dimensionality of the restricted subspace W is also extended to higher dimensions accordingly.