Help with PseudoInverse

Question

I am trying to (pseudo)solve a linear system $Ax=y$. I have a matrix A, and 2 vectors x and y

A = {
{1, 1, 1, 0},
{1, 1, 0, 1},
{0, 1, 1, 0},
{0, 1, 0, 1}};
x = {pcv, pcf, pat, pnt};
y = {p11, p00, p10, p01};

Reading Wikipedia, I expect to obtain A.PseudoInverse[A].y == y. However I obtain

In[4]:= A.PseudoInverse[A].y

Out[4]= {p00/4 - p01/4 + p10/4 + (3 p11)/4, (3 p00)/4 + p01/4 - p10/4 + p11/4, 
         -(p00/4) + p01/4 + (3 p10)/4 + p11/4, p00/4 + (3 p01)/4 + p10/4 - p11/4}

What am I missing?

Answer 1

The Moore-Penrose pseudoinverse of A is a right inverse only if A is surjective. But your A is not surjective since Transpose[A] has a nontrivial kernel:

NullSpace[Transpose[A]]

{{1,-1,-1,1}}

But as generalized inverse, you have of course

A.PseudoInverse[A].A == A
PseudoInverse[A].A.PseudoInverse[A] == PseudoInverse[A]

True

True

Addendum

Actually, this is true for each generalized inverses $B$ of $A$: If $y$ is not in the image of $A$, then $A \, B \, y$ cannot equal $y$ (the former lies in the image, the latter does not). But if $y$ is in the image of $A$, e.g., $y = A\,x$ then $A \, B \, y = A\, B\, A\, x = A \, x = y$. So we get:

$$A \, B\, y = y \quad \text{if and only if} \quad y \in \operatorname{ima}(A).$$