Linear Algebra and Its Applications · Problem 7.4.7

Linear Programming and Game Theory — Gilbert Strang

Mathematics · jacobi-gauss-seidel-2x2

Problem

For a general 2 by 2 matrix

A = \begin{bmatrix} a & b \\ c & d \end{bmatrix},

find the Jacobi iteration matrix

S^{-1}T = -D^{-1}(L+U)

and its eigenvalues

\mu_i

. Find also the Gauss-Seidel matrix

-(D+L)^{-1}U

and its eigenvalues

\lambda_i

, and decide whether

\lambda_{\max} = \mu_{\max}^2