Application: Simplification of quadratic forms

Which of the following are quadratic forms?

1. $f_{1}(x,y,z)=(x+y+z)^{2}$.

2. $f_{2}(x,y,z)=x^{2}+y^{2}+z^{2}$.

3. $f_{3}(x,y,z)=(x+y)^{2}+(x+z)^{2}+(y+z)^{2}$.

4. $f_{4}(x,y,z)=x^{2}-y^{2}$.

5. $f_{5}(x,y,z)=x^{2}+2xyz+(y+z)^{2}$.

6. $f_{6}(x,y,z)=x(y+z)$.

7. $f_{7}(x,y,z)=x^{2}y^{2}z^{2}$.

Write the quadratic form

in matrix form.

Apply the change of variables $x=u+2w$, $y=v$, $z=w-v$ to the quadratic form

Perform a change of variables so that the quadratic form $f(x,y)=3x^{2}-2xy+3y^{2}$ becomes diagonal.

Find the principal axes of the ellipsoid $2x^{2}+2y^{2}+3z^{2}+2xz-2yz$.