Kunze Linear Algebra · Problem 10.2.8

Bilinear Forms

Mathematics — Methods · diagonalizing-quadratic-form-rn

Problem

Let AA be a symmetric n×nn \times n matrix over RR, and let qq be the quadratic form on RnR^n given by q(x1,…,xn)=∑i,jAijxixj.q(x_1, \ldots, x_n) = \sum_{i,j} A_{ij} x_i x_j. Generalize the method used in Exercise 7 to show that there is an invertible linear operator UU on RnR^n such that (U†q)(x1,…,xn)=∑i=1ncixi2(U^\dagger q)(x_1, \ldots, x_n) = \sum_{i=1}^n c_i x_i^2 where cic_i is 11, −1-1, or 00, i=1,…,ni = 1, \ldots, n.

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