Linear Algebra and Its Applications · Problem 6.1.7

Positive Definite Matrices — Gilbert Strang

Mathematics · positive-definite-matrices-minima-maxima-saddle-points

Problem

(a) What 3 by 3 symmetric matrices

A_1

and

A_2

correspond to

f_1

and

f_2

\begin{aligned} f_1 &= x_1^2 + x_2^2 + x_3^2 - 2x_1 x_2 - 2x_1 x_3 + 2x_2 x_3 \\ f_2 &= x_1^2 + 2x_2^2 + 11x_3^2 - 2x_1 x_2 - 2x_1 x_3 - 4x_2 x_3. \end{aligned}

(b) Show that

f_1

is a single perfect square and not positive definite. Where is

f_1

equal to

0

? (c) Factor

A_2

into

LL^T

. Write

f_2 = x^T A_2 x

as a sum of three squares.