Singular value decomposition

Singular Value Decomposition (SVD) is widely used in linear algebra and is known for its strength, particularly arising from the fact that every matrix has an SVD. It looks like this:

For our purposes, let's suppose , , , and , and that U, V are orthogonal matrices, whereas ∑ is a matrix that contains singular values (denoted by σ_i) of A along the diagonal. 

∑ in the preceding equation looks like this:

We can also write the SVD like so:

Here, u_i, v_i are the column vectors of U, V.