Definition:
G is a generalized inverse of A if and only if AGA=A.G is said to be reflexive if and only if GAG=G.
I was trying to solve the problem:
If A is a matrix and G be it's generalized inverse then G is reflexive if and only if rank(A)=rank(G).
One direction of the proof is easy but I am finding it difficult to prove the second part.I have proved that GAG is infact a generalized inverse but why it should be equal to G but not any other generalized inverse is not very apparent to me
Best Answer
Suppose that $G$ is a generalized inverse (i.e. $AGA = A$) such that $A$ and $G$ have the same rank. Let $A = PQ$ be a rank decomposition, so that $P$ has $r$ columns and $Q$ has $r$ rows with $r$ equal to the rank of $A$. Similarly, let $G = BC$ denote a rank decomposition of $G$. We have $$ AGA = A \implies P(QB)(CP)Q = PQ. $$ Because $P$ has independent columns and $Q$ has independent rows, we can conclude from the above that $$ P[(QB)(CP) - I_r]Q = 0 \implies (QB)(CP) - I_r = 0 \implies (QB)(CP) = I_r, $$ which is to say that $(QB)^{-1} = (CP)$. With that, it follows that $$ GAG = B(CP)(QB)C = BC = G, $$ which is to say that $G$ is a reflexive inverse, which was what we wanted.