# Generalized inverse of a symmetric matrix

I have always found the common definition of the generalized inverse of a matrix quite unsatisfactory, because it is usually defined by a mere property, $A A^{-} A = A$, which does not really give intuition on when such a matrix exists or on how it can be constructed, etc… But recently, I came across a much more satisfactory definition for the case of symmetric (or more general, normal) matrices.