Linear Algebra and Its Applications · Problem 7.4.3

Linear Programming and Game Theory — Gilbert Strang

Mathematics · iterative-methods-jacobi-eigenvectors

Problem

In Problem 2, show that

x_k = (\sin k\pi h, \sin 2k\pi h, \ldots, \sin nk\pi h)

is an eigenvector of

A

. Multiply

x_k

A

to find the corresponding eigenvalue

\alpha_k

. Verify that in the 3 by 3 case these eigenvalues are

2-\sqrt{2}

2

2+\sqrt{2}

. Note. The eigenvalues of the Jacobi matrix

J = \frac{1}{2}(-L-U) = I - \frac{1}{2}A

are

\lambda_k = 1 - \frac{1}{2}\alpha_k = \cos k\pi h

. They occur in plus-minus pairs and

\lambda_{\max}

\cos\pi h