Kunze Linear Algebra · Problem 10.2.7

Bilinear Forms

Mathematics — Methods · diagonalizing-quadratic-form-r3

Problem

Let qq be the quadratic form on R3R^3 given by q(x1,x2,x3)=x1x2+2x1x3+x32.q(x_1, x_2, x_3) = x_1 x_2 + 2x_1 x_3 + x_3^2. Find an invertible linear operator UU on R3R^3 such that (U†q)(x1,x2,x3)=x12−x22+x32.(U^\dagger q)(x_1, x_2, x_3) = x_1^2 - x_2^2 + x_3^2. (Hint: Express UU as a product of operators similar to those used in Exercises 5 and 6.)

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