Answer to Question #275239 in Linear Algebra for Guhan

Question

Reduce 2++x²+2x₂x4x−2x4, into canonical form. Find the rank,

index, signature and its nature.

Answer 1

The reduction of a quadratic can be carried out by a procedure known as Lagrange’s Reduction, which consists essentially of repeated completing of the square:

$q=x_1^2+2x^2_2-7x_3^2-4x_1x_2+8x_1x_3=(x_1^2-4x_1(x_2-2x_3))+2x_2^2-7x_3^2=$

$=(x_1^2-4x_1(x_2-2x_3)+4(x_2-2x_3)^2)+2x_2^2-7x_3^2-4(x_2-2x_3)^2=$

$=(x_1-2x_2+4x_3)^2-2(x_2^2-8x_2x_3)-23x_3^2=$

$=(x_1-2x_2+4x_3)^2-2(x_2^2-8x_2x_3+16x_3^2)+9x_3^2=$

$=(x_1-2x_2+4x_3)^2-2(x_2-4x_3)^2+9x_3^2$

$y_1=x_1-2x_2+4x_3$

$y_2=x_2-4x_3$

$y_3=x_3$

$q=y_1^2-2y_2^2+9y_3^2$

index is  the number of positive terms in the canonical form:

$p=2$

the signature is number of positive terms minus the number of negative terms:

$n=2-1=1$

rank:

$r=n+p=3$

nature:

there is a mixture of positive and negative terms in the canonical form, so this is indefinite form