Getting ready for linear algebra exam. One question that I got right but not exactly sure why is this:(adsbygoogle = window.adsbygoogle || []).push({});

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Consider the quadratic form

Q(x,y,z) = 3x^2 + 3z^2 + 4xy + 4xy + 8xz

a) Decide if Q is positive definite, indefinite, etc.

b) What point on the surface Q = 1 lies closest to the origin and what is that distance?

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I computed the eigenvalues and got -1, -1 and 8, i.e. indefinite. But when just completing the square, there is only two positive terms: x(3x + 4y + 8z) + z(3z + 4y). How does this mesh with Sylvesters law of inertia?

Also, this form has got to be some kind of hyperboloid or something. So how can I know if the point associated with 1/sqrt(8) is actually on the surface? Since we're dealing with hyperbolas and not ellipses, that isn't always the case, is it?

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# Quadratic form, law of inertia

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