Let say I have 2 matrices (with a large dimensions), V, E. and $|V-E|_{2}\approx3\times10^{-11}$ I would expect that $VE^{+}=A$ would be very similar to the identity matrix up till replacement of some 1s in the main diagonals by 0s. Instead when I calculate it, I do get 1s in the main diagonals, and a lot of non-negligible "noise" in the other cells. in one of the cells of the main diagonal I actually get 0.43... I do understand that there are some counter-examples of matrices which exhibit some strange phenomena(as it's math) , but I would expect that almost all matrices would be "nice" in the sense they will be very close to the identity matrix (up to 0s switching 1s) when V is almost E. Is it related somehow to condition number?
What do I miss?