Linear Algebra and Its Applications · Problem 6.3.2

Positive Definite Matrices — Gilbert Strang

Mathematics · singular-value-decomposition

Problem

(a) Compute

AA^T

and its eigenvalues

\sigma_1^2

0

and unit eigenvectors

u_1

u_2

. (b) Choose signs so that

Av_1 = \sigma_1 u_1

and verify the SVD:

\begin{bmatrix} 1 & 4 \\ 2 & 8 \end{bmatrix} = \begin{bmatrix} u_1 & u_2 \end{bmatrix} \begin{bmatrix} \sigma_1 & \\ & 0 \end{bmatrix} \begin{bmatrix} v_1 & v_2 \end{bmatrix}^T.

C(A)

N(A)

C(A^T)

N(A^T)