# Is the unique least norm solution to $Ax=b$ the orthogonal projection of b onto $R(A)$?

True or false: The unique least norm solution to $Ax=b$ is the orthogonal projection of b onto $R(A)$

So, first isn't this the definition of least squares? The only incorrect thing I can think of is the solution may not be unique.

Also, the detailed answer states: The minimal solution of $Ax = b$ is in $R(A^T)$

Second, shouldn't it be in the $R(A)$, since it is the projection of $b$ onto the column space of $A$?

• I believe you are right, it is unique if $A$ is a full rank matrix. – Chee Han Oct 13 '16 at 15:29
• Suppose $A$ is $m \times n$. Then the least norm solution to $Ax = b$ belongs to $\mathbb R^n$, but the projection of $b$ onto $R(A)$ belongs to $\mathbb R^m$. So no, they are not equal. In fact they don't even belong to the same space. – littleO Aug 23 '20 at 23:11

## 1 Answer

Starting with $$\mathbf{A}x = b,$$ where the system matrix of rank $\rho$ has $m$ rows, and $n$ columns: $$\mathbf{A} \in \mathbb{C}^{m\times n}_{\rho}, \quad x \in \mathbb{C}^{n}, \quad b \in \mathbb{C}^{m}.$$ The general least squares problem is defined as $$x_{LS} = \left\{ x \in \mathbb{C}^{n} \colon \lVert \mathbf{A}x - b\rVert_{2}^{2} \text{ is minimized}\right\}$$ and the solution is $$x_{LS} = \color{blue}{\mathbf{A}^{\dagger} b} + \color{red}{\left( \mathbf{I}_{n} - \mathbf{A}^{\dagger}\mathbf{A} \right)y}, \quad y\in\mathbb{C}^{n}.$$ Range space components are in blue, nullspace in red.

The set of least squares minimizers is an affine space (dashed red line) passing through the range space of $\mathbf{A}$ at the point $x_{LS} = \mathbf{A}^{\dagger}b,$ as seen in the figure. Every point on the dashed line is a least squares minimizer. Which point has minimum length? That is, which point is closest to the origin? $$\lVert x_{LS}(y) \rVert_{2}^{2} = \lVert \color{blue}{\mathbf{A}^{\dagger} b} + \color{red}{\left( \mathbf{I}_{n} - \mathbf{A}^{\dagger}\mathbf{A} \right)y} \rVert_{2}^{2} = \lVert \color{blue}{\mathbf{A}^{\dagger} b} \rVert_{2}^{2} + \lVert \color{red}{\left( \mathbf{I}_{n} - \mathbf{A}^{\dagger}\mathbf{A} \right)y} \rVert_{2}^{2}$$ We can control the nullspace term in red by selecting the vector $y=0$; this choice is the least squares minimizer of minimum norm, $\color{blue}{\mathbf{A}^{\dagger} b}.$

Notice that the nullspace is trivial $\color{red}{\mathcal{N}\left( \mathbf{A} \right)}=\left\{ \mathbf{0}\right\}$ when $n=\rho$. Therefore, the set of minimizers is a point, and the solution is unique.