On the Eigenvector belonging to the Maximal Root of a Non-negative Matrix
Alexander M. Ostrowski · Proceedings of the Edinburgh Mathematical Society · 1960
By a theorem of Perron, a non-negative irreducible ( n × n ) matrix A = ( a μν ) has a positive fundamental root σ, the “ maximal root of A ”, such that the moduli of all other eigenvalues of A do not exceed σ. If we put σ lies between R and r . Since σ is not changed if A is transformed by a positive diagonal matrix D ( p 1 ,… p n , σ lies also between the expressions By a theorem of Frobenius, to σ as an eigenvalue of A belongs a positive eigenvector ξ, = ( x 1 …, x n ), satisfying