If A = ( a i j ) A = ({a_{ij}}) is an n × n n \times n irreducible matrix, then there are positive numbers d 1 , d 2 , ⋯ {d_1},{d_2}, \cdots , d n {d_n} so that ∑ k d i a i k d k − 1 = ∑ k d k a k i d i − 1 \sum \nolimits _k {{d_i}{a_{ik}}d_k^{ - 1} = } \sum \nolimits _k {{d_k}{a_{ki}}d_i^{ - 1}} for each i ∈ { 1 , 2 , ⋯ , n } i \in \{ 1,2, \cdots ,n\} . Further, the numbers d 1 , d 2 , ⋯ {d_1},{d_2}, \cdots , d n {d_n} are unique up to scalar multiples.