(a) Consider a system of linear equations with a non-singular square matrix . To determine its solution we apply the iterative method
Here , while the matrix is such that implies . The initial vector is arbitrary. Prove that, if the matrix possesses linearly independent eigenvectors whose corresponding eigenvalues satisfy , then the method converges for any choice of , i.e. as .
(b) Describe the Jacobi iteration method for solving . Show directly from the definition of the method that, if the matrix is strictly diagonally dominant by rows, i.e.
then the method converges.