Answer to Question #315114 in Linear Algebra for Beauty

Question #315114

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


1
Expert's answer
2022-03-22T18:50:35-0400

Q=x2+2y2+2z22xy2yz+zx==x2+2x(z2y)+(z2y)2(z2y)2+2y2+2z22yz==(x+z2y)2+y2yz+74z2==(x+z2y)2+(yz2)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) ^2

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



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