Answer to Question #315114 in Linear Algebra for Beauty

Question

Change Q= x² + 2y² + 2z² - 2xy - 2yz + zx into real canonical form and find its rank and signature.

Answer 1

$Q=x^2+2y^2+2z^2-2xy-2yz+zx=\\=x^2+2x\left( \frac{z}{2}-y \right) +\left( \frac{z}{2}-y \right) ^2-\left( \frac{z}{2}-y \right) ^2+2y^2+2z^2-2yz=\\=\left( x+\frac{z}{2}-y \right) ^2+y^2-yz+\frac{7}{4}z^2=\\=\left( x+\frac{z}{2}-y \right) ^2+\left( y-\frac{z}{2} \right) ^2+\left( \sqrt{\frac{3}{2}}z \right) ^2$

The rank is 3, the signature is $\left( 3,0 \right)$