Linear Algebra and Its Applications · Problem 7.4.18

Linear Programming and Game Theory — Gilbert Strang

Mathematics · arnoldi-gram-schmidt-krylov

Problem

In Arnoldi, show that

q_2

is orthogonal to

q_1

. The Arnoldi method is Gram-Schmidt orthogonalization applied to the Krylov matrix:

K_N = Q_N R_N

. The eigenvalues of

Q_N^T A Q_N

are often very close to those of

A

, even for

N \ll n

. The Lanczos iteration is Arnoldi for symmetric matrices (all coded in ARPACK).