## 每週問題 January 27, 2014

Let $A$ be an $m\times n$ real matrix and $\mathbf{b}\in\mathbb{R}^m$. Show that $\mathbf{z}$ is a least squares solution for $A\mathbf{x}=\mathbf{b}$ if and only if $\mathbf{z}$ is part of a solution to the larger system

$\displaystyle \begin{bmatrix} I_m&A\\ A^T&0 \end{bmatrix}\begin{bmatrix} \mathbf{y}\\ \mathbf{z} \end{bmatrix}=\begin{bmatrix} \mathbf{b}\\ \mathbf{0} \end{bmatrix}.$

$\displaystyle \begin{bmatrix} I_m&A\\ A^T&0 \end{bmatrix}\begin{bmatrix} \mathbf{y}\\ \mathbf{z} \end{bmatrix}=\begin{bmatrix} I_m&A\\ A^T&0 \end{bmatrix}\begin{bmatrix} \mathbf{b}-A\mathbf{z}\\ \mathbf{z} \end{bmatrix}=\begin{bmatrix} \mathbf{b}-A\mathbf{z}+A\mathbf{z}\\ A^T(\mathbf{b}-A\mathbf{z}) \end{bmatrix}=\begin{bmatrix} \mathbf{b}\\ \mathbf{0} \end{bmatrix}$

$\displaystyle \begin{bmatrix} I_m&A\\ A^T&0 \end{bmatrix}\begin{bmatrix} \mathbf{y}\\ \mathbf{z} \end{bmatrix}=\begin{bmatrix} \mathbf{b}\\ \mathbf{0} \end{bmatrix}$