Linear Algebra and Its Applications · Problem 7.2.4

Computations with Matrices — Gilbert Strang

Mathematics · matrix-norm-condition-number

Problem

For the positive definite

A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}

, compute

\|A^{-1}\| = 1/\lambda_1

\|A\| = \lambda_2

, and

c(A) = \lambda_2/\lambda_1

. Find a right-hand side

b

and a perturbation

\delta b

so that the error is the worst possible,

\|\delta x\|/\|x\| = c\|\delta b\|/\|b\|