Kunze Linear Algebra · Problem 10.2.5

Bilinear Forms

Mathematics — Methods · completing-square-quadratic-form

Problem

Let

q

be the quadratic form on

R^2

given by

q(x_1, x_2) = ax_1^2 + 2bx_1 x_2 + cx_2^2, \qquad a \neq 0.

Find an invertible linear operator

U

R^2

such that

(U^\dagger q)(x_1, x_2) = ax_1^2 + \left(c - \frac{b^2}{a}\right) x_2^2.

(Hint: To find

U^{-1}

(and hence

U

), complete the square. For the definition of

U^\dagger

, see part (e) of Exercise 4.)