How to prove a strictly diagonally dominant matrix is convergent

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SUMMARY

The discussion centers on proving the convergence of Jacobi's method for solving the linear system Ax = b when the coefficient matrix A is strictly diagonally dominant. The matrix A is expressed as A = D - L - U, where D is the diagonal matrix, -L is the strictly lower triangular part, and -U is the strictly upper part. The iteration matrix Tj = D-1(L + U) must be analyzed to show that its largest eigenvalue is less than 1, utilizing the Contraction Mapping Theorem as a key tool in the proof.

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  • Understanding of linear algebra concepts, particularly matrix decomposition.
  • Familiarity with Jacobi's method for iterative solutions of linear systems.
  • Knowledge of eigenvalues and eigenvectors in the context of matrix analysis.
  • Comprehension of the Contraction Mapping Theorem and its implications in iterative methods.
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  • Study the properties of strictly diagonally dominant matrices and their implications for convergence.
  • Learn about the derivation and application of the Contraction Mapping Theorem in numerical analysis.
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  • Investigate other iterative methods for solving linear systems, such as Gauss-Seidel and Successive Over-Relaxation (SOR).
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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?
 
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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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