Change Q= x² + 2y² + 2z² - 2xy - 2yz + zx into real canonical form and find its rank and signature.
Q=x2+2y2+2z2−2xy−2yz+zx==x2+2x(z2−y)+(z2−y)2−(z2−y)2+2y2+2z2−2yz==(x+z2−y)2+y2−yz+74z2==(x+z2−y)2+(y−z2)2+(32z)2Q=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) ^2Q=x2+2y2+2z2−2xy−2yz+zx==x2+2x(2z−y)+(2z−y)2−(2z−y)2+2y2+2z2−2yz==(x+2z−y)2+y2−yz+47z2==(x+2z−y)2+(y−2z)2+(23z)2
The rank is 3, the signature is (3,0)\left( 3,0 \right)(3,0)
Need a fast expert's response?
and get a quick answer at the best price
for any assignment or question with DETAILED EXPLANATIONS!
Comments
Leave a comment