Definition Let $A \in M_{m,n}(\mathbb{R})$ be a matrix. Then a matrix $B \in M_{n,m}(\mathbb{R})$ is called a pseudoinverse of $A$ if we have
- $ABA = A$ and
- $BAB = B$.
If in addition
- $AB$ and $BA$ are symmetric
then we call $B$ a Moore-Penrose pseudoinverse.
In the literature, one shows that every matrix $A \in M_{m,n}(\mathbb{R})$ has a unique Moore-Penrose pseudoinverse $A^+$.
My question: Does the uniqueness still hold if we omit condition 3., i.e. $B$ is a pseudoinverse which is not a Moore-Penrose pseudoinverse?