First,
(Q^T C Q)^T = Q^T C^T (Q^T)^T = Q^T C Q,
so Q^T C Q is symmetric.

Now,
det(Q^T C Q - xI)
= det(Q^T C Q - Q^T xI Q) . . . . [since Q^T Q = I]
= det(Q^T (C - xI) Q)
= det(Q^T) det(C - xI) det(Q)
= det(C - xI) det(Q^T) det(Q)
= det(C - xI) det(Q^T Q)
= det(C - xI) det(I) . . . . . . [since Q^T Q = I, again]
= det(C - xI)

Therefore the characteristic polynomials are the same, so in particular, the eigenvalues are the same. Therefore Q^TCQ is symmetric and every eigenvalue is positive real, so it is positive definite.

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