Unformatted text preview: • If A were irreducible then the eigenvalue at 1 would be simple; see, for example, [1]. • Let the eigensystem of A be deFned by Au j = λ j u j , and let e 1 = n X j =1 α j u j , where u 1 , . . . , u 4 are a basis for the eigenspace corresponding to the eigenvalue 1. Then verify that A k e 1 = n X j =1 α j λ k j u j . Since λ k j → 0 as k → ∞ except for the eigenvalue 1, we see that A k e 1 → α 1 u 1 + α 2 u 2 + α 3 u 3 + α 4 u 4 . • Therefore, we converge to a multiple of the stationary vector. [1] Richard Varga, Matrix Iterative Analysis , Prentice Hall, Englewood Cli±s, NJ, 1962....
View
Full Document
 Fall '11
 Dr.Robin
 Linear Algebra, Unit Circle, Gerschgorin Circle Theorem, Matrix Iterative Analysis, main diagonal element

Click to edit the document details