SVD and Pseudo-Inverse of a Matrix

The most general way for solving any linear system (consistent, overdetermined, or underdetermined) is to use the pseudo-inverse of the matrix. A consistent systan has a unique solution, an overdetermined system is one with more equations than unknowns, and an underdetermined system has an infinite number of solutions. If the pseudo-inverse is denoted by A, then the solution ofAx = b can be written as 

For a square, nonsingular matrix, A+ coincides with the inverse, A The pseudoinverse always exists, whether or not the matrix is square or has full rank. 

For an n X m matrix, A, A+ must satisfy the following four conditions  

The conditions that the pseudo-inverse must satisfy are not very helpful for computing A+. Computation is most easily accomplished by using the singular value decomposition (SVD) of A. The SVD can be visualized as a factoring of A into three separate matrices as follows  

Multiplying both sides of Equation 3.48 by and remembering that IF U = I, there results 

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