We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenva...We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .展开更多
The matrix equation A mn X ns B st =C over an arbitrary skew field is considered. A necessary and sufficient condition for the consistency and the expression for general solutions of the above mentioned matrix equatio...The matrix equation A mn X ns B st =C over an arbitrary skew field is considered. A necessary and sufficient condition for the consistency and the expression for general solutions of the above mentioned matrix equation are presented.Moreover,a practical method of solving one is also given.展开更多
文摘We determine the left eigenvector of a stochastic matrix M associated to the eigenvalue 1 in the commutative and the noncommutative cases. In the commutative case, we see that the eigenvector associated to the eigenvalue 0 is (N1,Nn) , where Ni is the i–th iprincipal minor of N=M–In , where In is the identity matrix of dimension n. In the noncommutative case, this eigenvector is (P1-1,Pn-1) , where Pi is the sum in Q《αij》 of the corresponding labels of nonempty paths starting from i and not passing through i in the complete directed graph associated to M .
文摘The matrix equation A mn X ns B st =C over an arbitrary skew field is considered. A necessary and sufficient condition for the consistency and the expression for general solutions of the above mentioned matrix equation are presented.Moreover,a practical method of solving one is also given.
