Linear Algebra and Its Applications · Problem 6.1.11

Positive Definite Matrices — Gilbert Strang

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

Problem

(a) If

A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}

is Hermitian (complex

b

), find its pivots and determinant. (b) Complete the square for

x^H A x

. Now

x^H = [\bar{x}_1\ \bar{x}_2]

can be complex:

a|x_1|^2 + 2\mathrm{Re}\, b\bar{x}_1 x_2 + c|x_2|^2 = a|x_1 + (b/a)x_2|^2 + \underline{\quad}|x_2|^2.

a > 0

and

ac > |b|^2

ensure that

A

is positive definite. (d) Are the matrices

\begin{bmatrix} 1 & 1+i \\ 1-i & 2 \end{bmatrix}

and

\begin{bmatrix} 3 & 4+i \\ 4-i & 6 \end{bmatrix}

positive definite?