Singular value decomposition (SVD) is a method for finding the equivalent of eigenvalues and eigenvectors for a rectangular matrix. It is what we use when we want to know the rank of a rectangular matrix, or make a low-rank matrix factorization (indeed, it is precisely what is used in principal components analysis or PCA).
The cool thing is: The method is exceedingly general; it can find the rank or a low-rank approximation to any space; it doesn't have to be a vector space exactly. It just has to obey certain algebra rules. So in my work with Soledad Villar (JHU) we use it to find a basis to represent all the linearly independent geometric images (grids of tensors) possible subject to constraints (like symmetries). I wrote words about using the SVD in this context in our nascent paper. Here is some example output of my SVDs in eye-candy form:
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