[Math] Intuitive explanation of left- and right-inverse

inverselinear algebra

I am reading about right-inverse and left-inverse matrices. According to theory if a matrix $A_{m\times n}(\mathbb{R})$ is full row rank, then it has a right-inverse. That is, $AC=I_{m}$. Similarly, if $A$ is full collumn rank, then it has a left-inverse. That is, $BA=I_{n}$. I have the following questions:

  1. Taking $AC=I_{m}\iff A^TAC=A^T \iff C=(A^TA)^{-1}A^T$ but this satisfies $CA=I$, contradiction. Similarly, taking $BA=I_{n}\iff BAA^T=A^T \iff B=A^T(AA^T)^{-1}$ but this satisfies $AB=I$, contradiction. How is that possible?

  2. Moreover, and most importantly what is the intuitive explanation of the left and right inverse? Is there any connection with the rows or collumns or any of the four foundamental subspaces of $A$?

Thank you very match!

Best Answer

  1. $A^\top A$ is singular when $A$ has full row rank, and more columns than rows. So, there is no inverse in the case. But $AA^\top$ is ok...

  2. one sided inverses reflect these properties of linear maps:

    • $A$ is surjective iff there is $B$ such that $AB=I$
    • $A$ is injective iff there is $B$ such that $BA=I$

Since linear maps can be of injective/surjective independently, this makes one sided inverses natural.

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