Solving a System of Linear Equations Using Pseudoinverse

Question

From what I know, Moore-Penrose Pseudoinverse is a method to solve overdetermined system of equations. Let $\mathbb{A}$ denote Pseudoinverse of $A$. I have the following matrices and I need to solve $Ax = b$, to do so I compute $x$ as $\mathbb{A}b$.

A =[
 4     2
-9    -3
 2     2
36     6]

and

b=[
 -53.5461
 -44.9373
 -51.7330
 -125.2530
 ]

the command x=pinv(A)*b in Matlab generates

x =
 -2.5988
 -3.1051

while Ax equals

-16.6054
 32.7045
-11.4079
-112.1873

and it is supposed to be close to b, but it is not at all. I would like to know what is wrong here. Thanks.

Answer 1

The values are correct.

Note that the system is overdetermined, which means there is no solution, and A*(pinv(A)*b) is giving you the projection of $b$ on the subspace generated by the columns of $A$ ($AA^+$ is the matrix of the orthogonal projection on the linear span of the columns of $A$). There is no reason it should be close to $b$.

Another way to see this: $x$ is exactly the vector of coefficients in the linear regression where the dependent variable is $b$ and the regressors are the columns of $A$. The difference between $Ax$ and $b$ is the vector of residuals.