Least square method solution for constant

Question

I would like to ask, why method of least square approximation (A*x=b) gives us solution, that is equal 1/n, where n is number of unknown elements.

example: Let A = $\begin{bmatrix} 7 & 7 & 7\\ 7 & 7 & 7 \\ 7 & 7 & 7\\ 7 & 7 & 7\end{bmatrix}$ and b=$\begin{pmatrix} 7\\7 \\7\\7\end{pmatrix}$ Why are then components of vector x equal to 0.33 (1/3). I understand that this would be minimal squared distances between points and new function, but I don't know how to give proof for it (that components of vector x are equal and they are x equal to 1/(number of components)

Answer 1

In this case $b\in col(A)$ thus the solution is trivial and all the solution $x=(a,b,c)$ with $a+b+c=1$ are valid.

Answer 2

The method of ordinary least squares will not yield the solution $x=(\frac13, \frac13, \frac13)$ as the system is under-determinate.

This solution will be obtained from the pseudo-inverse of the matrix, a more powerful tool. It selects the minimum norm solution among all those possible.

See https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse