# left inverse == right inverse with Moore–Penrose pseudoinverse?

1) No matter how the pseudoinverse is constructed, it is always the same? No matter if i use QR or SVD

2) Is the left inverse and the right inverse the same in Moore-Penrose pseudoinverses?

• There are in general many left-, right- and pseudoinverses, Moore-Penrose is just a particular construction with en extra nice property. – A.Γ. Jul 16 '15 at 12:16
• Yes, thats how i understood the generalized pseudoinverse. Thats why my question aims not for the general answer, but for the moore-penrose answer only... – helt Jul 16 '15 at 12:21
• What forms does the Moore-Penrose inverse take under systems with full rank, full column rank, and full row rank? math.stackexchange.com/questions/1537880/… – dantopa Mar 23 '17 at 22:28

2. If a matrix is not square, it may be invertible at most from one side, i. e. a left and a right inverses cannot exist for a non-square matrix simultaneously. If the matrix is square and has a left and a right inverse, then they are equal $$A^{-L}=A^{-L}(AA^{-R})=(A^{-L}A)A^{-R}=A^{-R}.$$ Moreover, the one side invertible square matrix must be necessarily invertible. For example, for left invertible we have $$n=\text{rank}\, I=\text{rank}\,A^{-L}A\le\text{rank}\, A\le n\quad\Rightarrow\quad\text{rank}\,A=n$$ and the left inverse, the right inverse, the M.-P. pseudoinverse and the normal inverse are the same.