Thread: Eigenvalue question

Hi.

Why is it a stochastic matrix always has the eigenvalue 1? I understand that the absolute value of a stochastic matrix can not exceed 1 since the determinant of the matrix has to be 1 or less than 1. But why will one eigenvalue always be 1?

Originally Posted by krje1980

Hi.

Why is it a stochastic matrix always has the eigenvalue 1? I understand that the absolute value of a stochastic matrix can not exceed 1 since the determinant of the matrix has to be 1 or less than 1. But why will one eigenvalue always be 1?

The vector with each component equal to 1 is an eigenvector for the transpose of any stochastic matrix, with eigenvalue 1. But the eigenvalues of a matrix and its transpose are the same.

Ah, that makes sense. Cheers! Thanks a lot!