Let's say that $r=rank(A)$ where $A$ is a rectangular matrix, and $r'=rank(A^+)$ where $A^+$ is the generalized inverse of $A$.
[$A^+ = VS^+U^T$]
Is there any relationship between $r$ and $r'$? Like a specific equality or inequality? Or maybe a way to calculate $r'$ by knowing the value of $r$?
My guess was that they're equal but then based on the definition of $A^+$ we might have some zeroes after doing the multiplication.
Best Answer
Yes. Because the $U$ and $V$ in the singular value decomposition are unitary ( hence invertible), we have $rank(A) = rank(S) = rank(S^+) = rank(A^+)$.
Multiplying by invertible matrices on either side does not influence rank.