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

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Answer 1 · 2018-02-01T09:09:55

#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.