Take the pseudoinverse of non-invertible and non-square matrices using singular value decomposition (SVD)

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A generalized process for finding the inverse of any matrix (including non-invertible and non-square matrices) uses SVD as follows:

Given matrix M:

\text{svd}(M) = U \cdot \Sigma \cdot V^T M^{-1} = V \cdot \Sigma^{-1} \cdot U^T

In pseudo-code:

const { U, S, VT } = svd(M)
const SInv = inv(S)
const MInv = dot(dot(VT.T, SInv), U.T)

(Taking the inverse of S is easy. All of the zeros remain zeros, and all the other values become their multiplicative inverse. For example, if:

S = [[3, 0], [0, 2]]

Then:

SInv = [[1/3, 0], [1/2, 0]]