Theorem states that for every initial value fixed point iteration x = Bx+b converges to the only solution of the system if all $|\lambda| $ < 1. Prove it using Jordans normal form. Initial form is Ax=b and to get B from A us whatever you want(Althoug nothing is known about A). Only thing is that if all eigenvalues of B are less than 1 then there is only 1 solution and fixed point iteration will find it.
- How can I represent the matrix B using Jordans normal form?
- What does Jordans normal form say about eigenvalues?
- Is there anything else i need to know to prove this?
I understand that Jordan normal form consists of Jordan boxes where diagonal has eigenvalue and diagonal on top has 1-s. But what determines the Jordans box size?