For matrices of less than full rank and arbitrary shape, the matrix inverse does not exist. The pseudoinverse of A is a unique matrix X in Rn x m that satisfies

(AX)T = AX
(XA)T = XA
The pseudoinverse of a matrix can be computed using its singular value decomposition. Using the SVD, an m×n matrix A can be expressed as
Therefore, using the properties of orthogonal matricies and the matrix inverse, the inverse of A is given by:
A -1 = (VT)-1Σ-1U -1 = VΣ-1UT
Of course, the formula above only works if Σ has an inverse. Σ may fail to have an inverse if it is singular or non-square.

To deal with this situation, we define a new n×m matrix called Σ by the equation

Σ = Diagn×m(1/σ1, 1/σ2, ..., 1/σr , 0, ..., 0)
where r is the rank of A. In other words, Σ is an n×m diagonal matrix which has the reciprocals of the non-zero singular values of A along its diagonal.

We then define the pseudoinverse of A as the n×m matrix A which is given by the equation

It is easy to show that this definition satisfies the axioms given in the previous writeup. To see this, first note that
ΣΣ = Diagn×n(1, 1, ..., 1, 0, ..., 0)
ΣΣ = Diagm×m(1, 1, ..., 1, 0, ..., 0)
where both matricies contain r 1's along their diagonals. Then use the relevant definitions.

The pseudoinverse of a square, non-singular matrix is equal to (surprise!) its inverse.

You can compute the minimum-norm least-squares solution to the system of linear equations Ax = b using the pseudoinverse. In other words, the column vector x with the shortest length which minimizes the expression

|| Ax - b ||
is denoted x* and is given by
x* = Ab.

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