One user asked me about the differences of SVD in Euler and Maxima. For quadratic matrices, both systems return orthogonal matrices U, V, and the diagonal d of a diagonal matrix D, such that

\(U\cdot D \cdot V = A\)

In fact, Maxima returns V‘. Maxima uses the LAPACK to compute this.

An obvious discrepancy might be the order of the singular values. But for non-quadratic nxm matrices, Maxima expands U to a square nxn orthogonal matrix, while Euler returns an nxm matrix with orthogonal columns. Consequently, in Maxima D in the above formula has to be expanded  with zeros to a nxm matrix.

If we want to get the result of Maxima in Euler, we need to expand a matrix with orthogonal vectors. There are several ways to achieve this. A stable method is to use svdkernel or orthogonal. The tricks are very simple.

>A = A | svdkernel(A');
>A = A | orthogonal(kernel(A'));

A notebook, which demonstrates an example can be found here.