Can a symmetric matrix AA^T be computed using matrix-vector operation?

I am interested in computing which is a symmetric matrix where now there's nothing special about A and in the worst case all of its entries are non-zero. I have an idea in my mind to calculate using level-2 BLAS operation.

First the diagonal entries of C is nothing but the j-th row of A multiplied by the transpose of the j-th row of A so it should be like this:

C(j,j)=A(j,:)*A(j,:)';

Next, I will compute the last entries of C that is to say I will compute the lower triangular part of C afterwards I will say that the upper triangular part is equal to the lower triangular part due to symmetry. And this is where my question arises. Can I compute the lower triangular part using matrix-vector operation while avoiding unecessary multiplication such as multiplying elements located at the upper triangular of A?

From what I reached I have reached the following expression :

C(j+1:n,j)=A(j+1:n,j)*A(:,j);
C(j,j+1:n)=C(j+1:n,j);

The issue is that this is a vector-vector multiplication (level-1 BLAS). I would hope for some context on whether level-2 BLAS is possible

[Matt J]

Would it be naive to ask why you don't simply do,

C=A*A.'

explicitly?

[Tarek Hajj Shehadi]

Hello Matt, the reason why I can't use any extremely optimized MATLAB built in function is because the course I am taking does not permit me to do so :( I have to wait a couple of months and then screw every non-effeciant algorithm I ever wrote!