CODEWITHSUNDEEP
matlablinear-algebra
Arnesh Nagavalli
Arnesh Nagavalli

Reputation: 51

Trouble creating an orthonormal matrix with given vector in matlab

Similar problem to this question matrix that forms an orthogonal basis with a given vector but I'm having problems with my implementation. I have a given vector, u, and I want to construct an arbitrary orthonormal matrix V such that V(:,1) = u/norm(u). The problem I'm getting is that V'*V isn't I (but is diagonal). Here's my code:

function V = rand_orth(u)
    n = length(u);
    V = [u/norm(u) rand(n,n-1)];

    for i = 1:n
        vi = V(:,i);
        rii = norm(vi);
        qi = vi/rii;
        for j = i+1:n
            rij = dot(qi,V(:,j));
            V(:,j) = V(:,j) - rij*qi;
        end
    end
end

Upvotes: 1

Views: 65

Answers (1)

nirvana-msu
nirvana-msu

Reputation: 4077

Simply normalize columns of V once you've finished the orthogonalization. Note there's also no need to normalize u (first column) initially, since we will normalize the whole matrix at the end:

function V = rand_orth(u)
    n = length(u);
    V = [u, rand(n,n-1)];

    for i = 1:n
        vi = V(:,i);
        rii = norm(vi);
        qi = vi/rii;
        for j = i+1:n
            rij = dot(qi,V(:,j));
            V(:,j) = V(:,j) - rij*qi;
        end
    end

    V = bsxfun(@rdivide, V, sqrt(diag(V'*V))');
end

Upvotes: 1

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