Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For $C$ a matrix, let $C^+$ be the pseudo inverse (Moore-Penrose inverse).

Let $B$ a square matrix ($n \times n$) with an inverse, and $A$ some matrix ($n \times m$). Under what conditions is it true that

$B^{-1} = A(BA)^+$ ?

(maybe some conditions on the rank of $A$ or something similar?)

share|cite|improve this question
up vote 4 down vote accepted

$B^{-1}=A(BA)^+$ if and only if $BA(BA)^+=I$. Hence the necessary and sufficient condition is $\textrm{rank}(BA)=n\le m$, i.e. $A$ is a full-rank "wide" matrix.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.