How to prove a strictly diagonally dominant matrix is convergent

  • #1
Question:

Ax=b

Let the coefficient matrix A be written in the form A=D-L-U, where D is the diagonal matrix whose diagonal is the same as that of A, -L is the strictly lower triangular part of A and -U is the strictly upper part of A. Furthermore, let Tj = D-1(L+U) be the iteration matrix for Jacobi's method. Prove that Jacobi's method is convergent if the coefficient matrix is diagonally dominant.

If A and b are given, I know how to use the Jacobi's method to find out whether or not A is convergent. But how should I prove that "Jacobi's method is convergent if A is diagonally dominant" using just those given letters and symbols?
 

Answers and Replies

  • #2
71
7
Here's a hint: Contraction Mapping Theorem.

(you want to show that the largest eigenvalue of the matrix D^-1x(L + U) is less than 1, then you can use the contraction mapping from there).
 

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