Answer to Question #146167 in Linear Algebra for Harish

Question

Reduce the quadratic form Q=x1²+2x2x3 to canonical form and hence find its nature, rank, index and signature.

Answer 1

$Q=x^2_1+2x_2x_3$

Let

$x_1 =y_1\\ x_2=y_2-y_3\\ x_3=y_2+y_3$

$Q=y^2_1+2(y_2-y_3)(y_2+y_3)=\\ =y^2_1+2y^2_2-2y^2_3$

canonical form is $Q=y^2_1+2y^2_2-2y^2_3$

rank=3 (№ of non-zero eigen values)

index= 2 (№ of positive eigen values)

signature=2-1=1 (difference betwen № of positive and negative eigen values)

nature: indefinite if some of the eigen values of Q are + ve and others – ve.