Vtyjkyuk

While trying to generalize a certain formula from the scalar case to the matrix case, I came across the following curious observation:

Let $M$ be a real, not necessarily square matrix, then
$$left(I - M M^T right)^-1 M = Mleft(I - M^T M right)^-1$$
which seems to be true in general. It holds, for example, for this randomly chosen case
$$M=left(
beginarrayccc
-1 & 7 & -4 \
-1 & 5 & 7 \
-4 & 7 & 4 \
-2 & -5 & 0 \
2 & -1 & -6 \
endarray
right)$$

Although numerical examples convinced me that this is true, I do not understand why. Any hints?

More in detail we have that if $M$ is a n-by-m matrix

• $X=I - MM^T$ is a n-by-n matrix




• $Y=I - M^T M$ is a m-by-m matrix

and therefore

$$X_ntimes n^-1M_ntimes m=M_ntimes mY_mtimes m^-1$$

$$X_ntimes nX_ntimes n^-1M_ntimes mY_mtimes m=X_ntimes nM_ntimes mY_mtimes m^-1Y_mtimes m$$

$$I_ntimes nM_ntimes mY_mtimes m=X_ntimes nM_ntimes mI_mtimes m$$

$$M_ntimes mY_mtimes m=X_ntimes nM_ntimes m$$

that is

$$M left(I - M^T M right) = left(I - M M^T right) M$$

Thanks to @Empy2, just multiply both sides with the "denominator" from the other side:
$$left(I - M M^T right)^-1 M = Mleft(I - M^T M right)^-1$$
$$iff M left(I - M^T M right) = left(I - M M^T right) M$$
$$iff M - M M^T M = M - M M^T M$$