Application Of The Weyr Canonical Form In A Class Of Matrix Equations

The theory and application of Weyr canonical form have been studied systematically and in depth.The Weyr canonical form has a wide range of applications.We use the Weyr canonical form to study an important class of linear transformations.Let φAB be a linear transformation of the m×n-dimensional vector space Mm×n(C)over the complex field C such that φAB(X)=AX-XB,where A and B m×m and n×n complex matrices,respectively.φAB is a class of very fundamental linear transformations,especially in the study of Lie algebra.ker φAA is the solution space of the matrix equation AX-XA=0.The dimension formula for the kernal ker φA is due to Frobenius.Subsequently,the dimensional formulas of the kernels of the linear transformations φA2 and φA3 are given,which generalizes the work of Gracia.And recently Liao Jun et al.use the Jordan canonical form to obtain the basis and dimensional formulas for kernels of each power of the transformation φABk(k>1).This paper uses the Weyr canonical form to solve the dimension of ker φABk when k is small,and reduces the problem to the case of single blocks.The matrix Xthat satisfies the solution space conditions of the the matrix equation are given by calculating matrix equation,using the elementary transformation of matrix and the operation of block matrix.And then we obtain the basis and dimension formulas of ker φWA(γ)WB(γ)k.Finally,the dimensional formulas of ker φABk are given(k=1,2,3,4).