Linear Algebra and Its Applications · Problem 6.1.2

Positive Definite Matrices — Gilbert Strang

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

Problem

Decide for or against the positive definiteness of these matrices, and write out the corresponding

f = x^T A x

: (a)

\begin{bmatrix} 1 & 3 \\ 3 & 5 \end{bmatrix}

. (b)

\begin{bmatrix} 1 & -1 \\ -1 & 1 \end{bmatrix}

. (c)

\begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}

. (d)

\begin{bmatrix} -1 & 2 \\ 2 & -8 \end{bmatrix}

. The determinant in (b) is zero; along what line is

f(x,y) = 0