It would be interesting to see how the positive definite is known. Page https://en.wikipedia.org/wiki/Positive-definite_matrix gives an example for which v.T*M*v gives 2*a**2 - 2*a*b + 2*b**2 - 2*b*c + 2*c**2 which is factored into a sum of squares (and hence nonnegative). I don't know if we have a way of factoring such expressions into squares, however.
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