Linear Algebra and Its Applications · Problem 11

Positive Definite Matrices — Gilbert Strang

Mathematics · singular-value-decomposition

Problem

Suppose

A

is invertible (with

\sigma_1 > \sigma_2 > 0

). Change

A

by as small a matrix as possible to produce a singular matrix

A_0

. Hint:

U

and

V

do not change: Find

A_0

from

A = \begin{bmatrix} u_1 & u_2 \end{bmatrix} \begin{bmatrix} \sigma_1 & \\ & \sigma_2 \end{bmatrix} \begin{bmatrix} v_1 & v_2 \end{bmatrix}^T