Linear Algebra and Its Applications · Problem 6.1.5

Positive Definite Matrices — Gilbert Strang

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

Problem

(a) For which numbers

b

is the matrix

A = \begin{bmatrix} 1 & b \\ b & 9 \end{bmatrix}

positive definite? (b) Factor

A = LDL^T

when

b

is in the range for positive definiteness. (c) Find the minimum value of

\frac{1}{2}(x^2 + 2bxy + 9y^2) - y

for

b

in this range. (d) What is the minimum if

b = 3