Matrices with more columns than rows

[photopea]

How can I compute a SVD decomposition of a matrix, which has more columns than rows?

I tried to transpose the matrix and switch the U and V results, but there are still some differences in signs (plus and minus).

[donhatch]

I tried to transpose the matrix and switch the U and V results, but there are still some differences in signs (plus and minus).

That should work.

Can you clarify what you mean by "there are still some differences in signs (plus and minus)"?
Note that the SVD isn't necessarily unique, so if you are comparing to some other calculated result, then you may see differences, which don't necessarily mean the answer is incorrect.

Are you seeing that the final UΣV^T is not equal to your input matrix?

[facelessuser]

Hopefully, this helps. Wide and tall matrices can be handled with small alterations.

A wide matrix can simply transpose the U and swap the m and n and pass it through the algorithm. At the end, you need to swap U and V. This will get you a reduced SVD. If you want the full matrix, you need to pad out U. V and Q need to be trimmed before returning the full matrix.

A Tall matrix will pass through fine and give you a reduced SVD, but you need to pad out U for a full matrix. V and Q need to be trimmed before returning the full matrix.

    eps = EPS
    tol = MIN_FLOAT / EPS

    u = acopy(a)
    square = m == n
    wide = not square and m < n
    diff = 0

    if wide:
        u = transpose(u)
        m, n = n, m

    if full_matrices and not square:
        diff = m - n
        for r in u:
            r.extend([0.0] * diff)
        n = m

    e = [0.0] * n
    q = [0.0] * n
    v = zeros((n, n))

    g, l, x, y = _householder_reduction_bidiagonal(m, n, e, u, q, tol)
    if compute_uv:
        g = _accumulate_right_transfrom(n, g, l, e, u, v)
        g = _accumulate_left_transform(m, n, g, l, u, q)
    _diagonalization_of_bidiagonal(m, n, g, x, y, e, u, q, v, eps)

    if full_matrices and not square:
        if compute_uv:
            v = [r[:-diff] for r in v[:-diff]]
        q = q[:-diff]

    if compute_uv:
        if wide:
            v, u = u, v

    if compute_uv:
        return u, q, v
    return q

To verify, we can give it tall matrix and test that $A = U \Sigma V^T$. We must disable full matrices to reconstruct the original matrix.

>>> r = alg.svd([[4, 5], [3, 3], [9, 7]], full_matrices=False)
>>> alg.pprint(alg.matmul(r[0], alg.matmul(alg.diag(r[1]), alg.transpose(r[2]))))
[[3.9999999999999987, 4.999999999999999],
 [3.0, 3.0000000000000004],
 [9.0, 7.0]]

And we can test wide:

>>> r = alg.svd([[4, 5, 3], [9, 7, 3]], full_matrices=False)
>>> alg.pprint(alg.matmul(r[0], alg.matmul(alg.diag(r[1]), alg.transpose(r[2]))))
[[4.0, 4.999999999999998, 2.999999999999999],
 [8.999999999999998, 6.999999999999996, 2.9999999999999982]]