Moore-Penrose Pseudo-Inverse Matrix

The classical inverse of a matrix is restricted to regular (square, non-singular) matrices. This restriction has severe consequences regarding the type of multivariate problems which can be solved by matrix algebra. In order to create the necessary tools for extending the range of solvable problems, the idea of inverting a given matrix has been extended to a more general level:
 

Moore-Penrose Pseudo-Inverse Matrix

Let A be an arbitrary matrix of order mn, and B be a matrix of order nm. B is called the "Moore-Penrose pseudo-inverse" of A, if ABA = A BAB = B AB is symmetric BA is symmetric The Moore-Penrose pseudo-inverse of A is usually depicted by A+.

What does this strange definition mean?  Simply stated, the first two statements mean, that we neglect all non-invertible properties of a matrix while inverting the rest. The other two statements choose a suitable matrix B of all those matrices that satisfy the first two rules.

If a system of linear equations is not solvable, one would like to know a good approximation of the solution anyway. More exactly expressed, one would like to find a solution which minimizes the error - and the Moore-Penrose pseudo-inverse really gives the best approximation.