Are all the conditions of the Moore-Penrose inverse definitions necessary?

Question

The Moore-Penrose inverse of a real or complex matrix $M$ is the unique matrix defined by four conditions. Can any of these four conditions be relaxed with no loss of uniqueness? I noticed there was a similar question to mine a while ago, but the question wasn't answered there.

Answer 1

No. Let $A=\pmatrix{1&0\\ 0&0},B_1=\pmatrix{x&0\\ 0&0},B_2=\pmatrix{1&0\\ 0&x},B_3=\pmatrix{1&x\\ 0&0},B_4=B_3^T$ where $x$ is arbitrary. Then

• $B=B_1$ satisfies the conditions that $BAB=B$ and that $AB$ and $BA$ are Hermitian.
• $B=B_2$ satisfies the conditions that $ABA=A$ and that $AB$ and $BA$ are Hermitian.
• $B=B_3$ satisfies the conditions that $ABA=A,BAB=B$ and that $BA$ is Hermitian.
• $B=B_4$ satisfies the conditions that $ABA=A,BAB=B$ and that $AB$ is Hermitian.