I took a linear algebra class, and the professor said that if you multiply the left-inverse to the right-side of the matrix, the result is a zeros added identity matrix as shown in figure 4.
- for left inverse, $A_{m \times n}, m>n$
$A^{-1}_{n \times m}A_{m \times n} = I_{n \times n}$ (figure 3)
$A_{m \times n}A^{-1}_{n \times m} = B_{m \times m}$ where $B_{m \times m}$ is a zeros added identity matrix (please refer to the figure 4)
- for right inverse, $A_{m \times n}, m<n$
$A_{m \times n} A^{-1}_{n \times m} = I_{m \times m}$ (figure 1)
$A^{-1}_{n \times m} A_{m \times n} = C_{n \times n}$ where $C_{n \times n}$ is a zeros added identity matrix (please refer to the figure 2)
Why would the result of multiplying the left inverse to the right be a zeros added identity matrix? Is there a book or web site covering this?
It isn't true that reversing the order of multiplication with a one-sided inverse results in such a matrix. For example, $\begin{bmatrix}1&1\end{bmatrix}\begin{bmatrix}1/2\\1/2\end{bmatrix}=\begin{bmatrix}1\end{bmatrix}$ so the left one is left inverse to the right.
But reversing them:
$\begin{bmatrix}1/2\\1/2\end{bmatrix}\begin{bmatrix}1&1\end{bmatrix}=\begin{bmatrix}1/2&1/2\\1/2&1/2\end{bmatrix}$
You would need some additional conditions on the left inverse for the statement you cited to hold.