Kunze Linear Algebra · Problem 8.1.8

Inner Product Spaces

Mathematics — Methods · inner-products

Problem

Let

A

be a

2 \times 2

matrix with real entries. For

X

Y

R^{2\times 1}

let

f_A(X, Y) = Y^t A X.

Show that

f_A

is an inner product on

R^{2\times 1}

if and only if

A = A^t

A_{11} > 0

A_{22} > 0

, and

\det A > 0