Question

Find the orthogonal canonical reduction of the quadratic form `x²+y²+z²-2xy-2xz-2yz`. Also find its principal axes?

Answer 1

`x²+y²+z²-2xy-2xz-2yz = p^top cdot M cdot p`

where

`p = (x,y,z)^top` and

`M = ((1,-1,-1),(-1,1,-1),(-1,-1,1))`

Now solving

`M p = lambda p` we obtain the eigenvalues and the eigenvectors

`V = ((-1,0,1),(-1,1,0),(1,1,1))`

and then

`M_c = V cdot M cdot V^-1 = ((2,0,0),(0,2,0),(0,0,-1))`

or calling now `P = (X,Y,Z)`

`P^top cdot M_c cdot P = 2X^2+2Y^2-Z^2`

which is the quadratic in the new coordinates system.

The classification is left as an exercise.