The singular values
They are the lengths of the ellipse's semi-axes, always non-negative, and conventionally listed largest first.
The largest is how much the matrix can stretch any vector — its gain in the worst case.
The smallest is how much it can squash one.
Their ratio is the condition number, and it is the single most useful number in the readout. A condition number near 1 means the matrix treats all directions roughly alike. A large one means it nearly flattens some direction, and solving anything involving that matrix will amplify error along it. Drag the entries until the two singular values are far apart and watch the ellipse become a sliver.
A zero singular value means the matrix genuinely flattens a direction: the ellipse degenerates to a line segment, the matrix is singular, and it has no inverse.
Why not eigenvectors
The [eigendecomposition](eigenvalues_and_eigenvectors.html) is the more famous factorisation, and the SVD is the more useful one, for three reasons.
It always exists. Eigendecomposition requires a square matrix, and even then not every square matrix has one. The SVD exists for every matrix of every shape, including a 1000×3.
Its bases are orthonormal. Eigenvectors need not be perpendicular, and when they are nearly parallel the decomposition is numerically fragile. U and V are [orthonormal](basis_span_and_orthogonality.html) by construction.
Its values are real and non-negative. Eigenvalues of a real matrix can be complex. Singular values never are.
For a symmetric positive-definite matrix the two coincide, which is why they are so often confused.
Low-rank approximation
The property that makes the SVD indispensable: keep only the largest *k* singular values, set the rest to zero, and multiply back. The result is provably the best rank-*k* approximation of the original matrix, in the least-squares sense. Not a good one — the best.
That single fact underwrites a surprising amount:
PCA is the SVD of the mean-centred data matrix. The principal components are the right singular vectors and the explained variances are the squared singular values, which is why any PCA implementation you look inside is calling an SVD.
Image compression keeps the top singular values of the pixel matrix.
Latent semantic analysis factors a term-document matrix and reads the retained dimensions as topics.
Recommender systems factor a sparse user-item matrix into a low-rank product.
Noise reduction works on the assumption that signal concentrates in the large singular values and noise spreads through the small ones.
Where it goes wrong
Forgetting to centre before PCA. Without subtracting the mean, the first component points at the mean rather than at the direction of greatest variance.
Reading a condition number as an error. It is a sensitivity: it says how much input error can be amplified, not how much has been.
Assuming singular values are eigenvalues. They are the eigenvalues of A'A, square-rooted. Not the same object, and not equal except in special cases.
Computing the full SVD when you want the top few. For a large sparse matrix a truncated solver is orders of magnitude cheaper.