Kunze Linear Algebra · Problem 5

Operators on Inner Product Spaces

Mathematics — Methods · positive-matrices

Problem

Let

A = \begin{bmatrix} 1 & \frac{1}{2} \\ \frac{1}{2} & \frac{3}{4} \end{bmatrix}.

(a) Show that

A

is positive. (b) Let

V

be the space of

2 \times 1

real matrices, with the inner product

(X|Y) = Y^t A X

. Find an orthonormal basis for

V

, by applying the Gram-Schmidt process to the basis

\{X_1, X_2\}

defined by

X_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad X_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.

2 \times 2

real matrix

P

such that

A = P^t P