Question

Prove that the quadratic form q(h) = h . Mh is indefinite if and only if M has both positive and negative eigenvalues.

Answer 1

Proof.

we prove it directly by using definition of indefinite form as folowing types:-
A quadratic form Q is indefinite if Q(x) assumes both positive and negative values

(B^TB)^T = B^T(B^T)^T = B^T B

then B^TB is a symmetric matrix.
Now, if it is positive (semi)definite. For all x, then
x^TB^TBx = (Bx)^T(Bx)

= (Bx) · (Bx) (by definition of inner product)

= ||Bx||² ≥ 0, (by definition of norm)

Now , by the definition, B^TB is positive semidefinite.

If B is invertible, for all x not equal to 0, we have Bx =not equal to 0, then x^TB^TBx = (Bx)T(Bx)

=(Bx) · (Bx)

= ||Bx||2 > 0, (by property of norm)

if x not equal to 0

Hence by definition, B^TB is positive definite

So we can conclude that Q(x) will be indefinite iff M has both positive and negative eigenvalues

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