Certificate: First of all, (a ta) t = a t (a t) t = a ta.

Therefore, A^TA is a symmetric matrix.

For any nonzero column vector x

According to r(A) = n, AX=0 has only zero solution.

So Ax ≠ 0

Then a is a real matrix,

So (ax) t (ax) > 0.

That is, x t (a ta) x >; 0

So A^TA is a positive definite matrix.