QR decomposition - Why is the first element of an orthogonal matrix always negative?

Hi, I noticed that the first element of an orthogonal matrix extracted from the qr() function is always negative, whatever the initial matrix and its size. The other elements do not have fixed signs though.

In the simple code below, j=0 always.

j=0;
for i=1:1:100000
    [Q,~] = qr(randn(3,3));
    if Q(1,1)>=0
        j=j+1;
    else
    end  
end  
disp(j)

Why is that so ?

Answer 1

[John D'Errico]

It is entirely arbitrary. If

Q*R = A

then it is equally true that

(-Q)*(-R) = A

Q is just a set of orthogonal vectors. They could have arbitrarily switched the signs on each vector in Q, which would have impacted R of course, but not in any meaningful way. And there is no constraint on the signs of the diagonal elements of R either. So it appears to have simply been an arbitrary choice made.

[Etienne Vaccaro-Grange]

I see, thanks for your reply !