I have the identity matrix $I_N\in\Re^{N\times N}$.
Now I consider matrix $A\in\Re^{N\times (N-1)}$ which is composed by a generic subset of of N-1 columns of matrix $I_N$.
I find out in my MATLAB code that this property always hold: pinv(A) is equal to $A^T$.
How can I prove that fact?
Best regards.
One proof is as follows. As is noted on the Wikipedia page for the pseudoinverse, if $A$ has linearly independent columns, then its pseudoinverse is given by $$ A^+ = (A^TA)^{-1}A^T. $$ On the other hand, if $A$ is formed using the columns of the identity matrix, then $A$ must have orthonormal columns, which means that $A^TA = I_{N-1}$. Conclude that $$ A^+ = I_{N-1}A^T = A^T, $$ which is what we wanted.