Sign differences in QR decomposition

Asked by Travis Kocian on 1 Aug 2013

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I am trying to use a simple QR decomposition and compare the results to that of the qr matlab function. I generate a random A matrix and then compare the Q and R values separately. For some reason, although the magnitudes of each element is the same there is sometimes a difference in sign between some of the terms. This is fine in the fact that Q*R = A in both cases, however I need to use R*Q. Any help on what is happening/how to fix it?

    m = 4;
    casenum = randi(3,1)
    if casenum == 1
        A = randn(m)
    end
    if casenum == 2
        A = i*randn(m)
    end
    if casenum == 3
        pownum = randi(2,m);
        A = randn(m) + randn(m).*(i.^pownum)
    end
    [Q1,R1] = qr(A);
Q2 = zeros(m,m);
R2 = zeros(m,m);
for i=1:m
    % Grab the next column of A.
    qbar = A(:,i);
    % Compute the current vector projection onto the previous columns of Q
    R2(:,i) = Q2'*qbar
    % Subtract out the projections so qbar is orthogonal
    qbar = qbar - Q2*R2(:,i);
    % Normalize qbar and append it to Q
    R2(i,i) = norm(qbar);
    Q2(:,i) = qbar/R2(i,i);
end

1 Comment

Jan on 2 Aug 2013

When Q*R=A is correct in both cases, both cases are correct results. This means unequivocally, that there is no "sign error" and I have deleted the corresponding tag. As Richard has explained already, the QR-decomposition is not unique. So you observe the expected effects.

1 Answer

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Answer by Richard Brown on 1 Aug 2013

Accepted Answer

It's only unique up to the signs of the rows of R. If you want to enforce positive diagonals of R, and thereby get a unique factorisation, construct

D = diag(sign(diag(R)));

and then

Qunique = Q*D; Runique = D*R;

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Travis Kocian (view profile)