We'll prove that A T .A is equal to its transpose. Prove that if A is skew-symmetric, then A.A is symmetric. A.A Hence, A.A is symmetric. All elements of the matrices are real numbers. Examine if X .
X T can be negative.
First we see that X . X T is a 1 x 1 matrix. It is a number, so maybe it is negative. But, X . X T can not be negative.