Pseudoinverse

  The inverse A^-1 of a matrix A exists only if A is square and has full rank. In this case, Ax = b has the solution x = A^-1b.

The pseudoinverse A⁺ is a generalization of the inverse, and exists for any (m,n) matrix. We assume m > n. If A has full rank (n) we define:

and the solution of Ax = b is x = A⁺b.

The best way to compute A⁺ is to use singular value decomposition. With , where U and V (n,n) are orthogonal and S (m,n) is diagonal with real, non-negative singular values s_i , we find

If the rank r of A is smaller than n, the inverse of does not exist, and one uses only the first r singular values; S then becomes an (r,r) matrix and U,V shrink accordingly. see also Linear Equations.