The author of the article studied the increasing character about the solutions of the matrix equation AX=0,with the character,the author gave out the method of solving power matrix equation A~mX=0,by increasing a maximum suitably increasing basic set of solutions of AX=0,and solved the difficulty in solving A~mX=0,if the power exponent m is biggish.

In this paper,the concept of nilpotent matrix is used to discuss some characters of the nilpotent matrix in general number field.

However, the properties of nilpotent matrix have not been much explored although its definition is given in discussing the multiplication of matrix.

This paper is derived to the study of the equation X~m=A where A is a n×n nilpotent matrix with 2≤m∈N.

In this paper,a necessary and sufficient conditions on the gcd closed set S with |S|=4 such that the power GCD matrix(Se)on S divides the power LCM matrix on S in the ring M4(Z) of 4×4 matrices over the integers is proved.

Shaofang Hong conjectured in 2002 that for a given positive integer t there is a positive integer k(t) depending only on t, such that if n≤k(t), then the power LCM matrix ([x_i, x_j]~t) defined on any gcd-closed set S={x_1,…,x_n} is nonsingular; but for n≥k(t)+1, there exists a gcd-closed set S={x_1,…,x_n} such that the power LCM matrix ([x_i, x_j]~t) on S is singular.

In this paper, we showthat for any real number e ≥1 and n ≤7, the power LCM matrix ([x_i,x_j]~e) definedon any gcd-closed set S = {x_1,.

On Nilpotent Matrices over Idempotent and Right-sided Quantale;

In this paper,we characterize isotropy subgroups of Jordan normal form of 3-nilpotent matrices under the conjugate action of GLn(F).

This paper devotes to an approach to nilpotent matrices and gives a classification theorem on lower dimensional nilpotent algebras.