GENERALIZED INVERSE
Generalized inverse - Suppose A is any m by n matrix. A+ is a generalized inverse of A if A+ is n by m and AA+A = A. Then, a fundamental theorem for linear equations is:
The equation, Ax = b, has a solution if, and only if, AA+b = b for any generalized inverse, A+, in which case the solutions are of the form: x = A+b + (I - A+A)y for any y in R+n.
Example . The
Moore-Penrose class additionally requires that AA
+ and A
+A be symmetric (hermitian, if A is complex). In particular, if rank(A) = m, A
+ = A
'Inv(AA
') is a Moore-Penrose inverse. Note that AA
+ = I, and A
+A is a projection matrix for the subspace (x: x = A
'y for some y in R
+m), since x = A
'y implies [A
+A]x = A
'Inv(AA
')AA
'y = A
'y = x. Further, I-A
+A projects into its orthogonal complement, which is the null space of A: A[(I-A
+A)x] = (A-A)x = 0.
