Linear Algebra and Its Applications · Problem 6.5.6

Computations with Matrices — Gilbert Strang

Mathematics · finite-element-method-subspace

Problem

A basic identity for quadratics shows

y = A^{-1}b

as minimizing:

P(y) = \frac{1}{2}y^TAy - y^Tb = \frac{1}{2}(y - A^{-1}b)^T A(y - A^{-1}b) - \frac{1}{2}b^T A^{-1}b.

The minimum over a subspace of trial functions is at the

y

nearest to

A^{-1}b

. (That makes the first term on the right as small as possible; it is the key to convergence of

U

u

.) If

A = I

and

b = (1,0,0)

, which multiple of

V = (1,1,1)

gives the smallest value of

P(y) = \frac{1}{2}y^Ty - y_1