We mark $A^+$ as a moore penrose inverse matrix
$A^+A$,$I-A^+A$,$I-AA^+$,$AA^+$
These four matrices are easy to verify that they are symmetric idempotent matrices, that is, orthogonal projection matrices, but to which spaces are they orthogonal projection matrices?
This is what I thought at first
- $A^+A$ is an orthogonal projection matrix to the row space of A
- $I-A^+A$ is an orthogonal projection matrix to the null space of A
- $AA^+$ is an orthogonal projection matrix to the column space of A
- $I-AA^+$ is an orthogonal projection matrix to the left null space of A
Because $AA^+Ax=Ax$ and $AA^+x=0\iff x'AA^+=0$，I think item 3 and item 4 are correct, but this is different from common projection matrix $A(A'A)^-A'$, so I think it is not quite right.
Then I can’t prove the first and second item.
Similarly,we marked $A^-$ as generalized inverse matrix if and only if $AA^-A=A$
$A^-A$，$I-A^-A$，$AA^-$，$I-AA^-$
These four matrices are easy to verify that they are idempotent matrices, that is, projection matrices, but to which spaces are they projection matrices?
This is what I thought at first
- $A^-A$ is an projection matrix to the row space of A
- $I-A^-A$ is an projection matrix to the null space of A
- $AA^-$ is an projection matrix to the column space of A
- $I-AA^-$ is an projection matrix to the left null space of A
But it's not right to think about it later, because the row space and the orthogonal complement of row space must be two orthogonal subspaces. It's impossible that the two projection matrices are not orthogonal and their projection spaces are orthogonal.
I hope you can help me. Thank you