Jacobi Method

Iterative Methods for Linear Algebraic Systems
Let where they are respectively: lower triangle, diagonal and upper triangle.
Set
We obtain the next iteration by solving

so

Theorem

If is Strictly diagonally dominant, then the Jacobi method converges.

Proof

Note so
Thus we need to prove
Let be an eigenvalue of .
Then

where the second line is obtained by multiplying by
Let
Suppose
Then is also strictly diagonally dominant.
So has evals with strictly positive real part by Gershgorin Theorem.
This is a contradiction, thus
So .

Theorem

If both and are symmetric positive definite, then Jacobi method converges

Proof

So setting we find
Now use The Householder-John Theorem to find