Linear Algebra and Its Applications · Problem 3.1.31

Orthogonality — Gilbert Strang

Mathematics · orthogonal-vectors-subspaces

Problem

Suppose

A

is a symmetric matrix (

A^T = A

). ((a)) Why is its column space perpendicular to its nullspace? ((b)) If

Ax = 0

and

Az = 5z

, which subspaces contain these ``eigenvectors''

x

and

z

? Symmetric matrices have perpendicular eigenvectors (see Section 5.5).