Prove Theorem 9.7.1.
Theorem 9.7.1
Principal Axes Theorem for ft3
Let
ax2 + by2 + cz2 + 2dxy + 2exz + 2fyz + gx + hy + iz + j = 0
be the equation of a quadric Q, and let
xTAx = ax2 + by2 + cz2 + 2dxy + 2exz + 2fyz
be the associated quadratic form. The coordinate axes can be rotated so that the equation of Q in the x′y′z′-coordinate system has the form
λ1x′2 + λ2y′2 + λ3z′2 + g′x′ + h′y′ + i′z′ + j = 0
Where λ1, λ2, and λ3 are the eigenvalues of A. The rotation can be accomplished by the substitution
x = Px′
Where P orthogonally diagonalizes A and det(x) = 1.