91Ó°ÊÓ

Consider a linear system as $\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$.

Here A is an matrix, a vector$\stackrel{→}{\mathrm{x}}$ in${\mathrm{R}}^{\mathrm{n}}$ called a least square solution of this system if$||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}{\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}||≤||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}}||$ for all$\stackrel{→}{\mathrm{x}}$ in ${\mathrm{R}}^{\mathrm{m}}$.

Consider a linear system as $\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$where $\left[\begin{array}{ccc}.& .& .\\ {\stackrel{→}{\mathrm{u}}}_1& ...& {\stackrel{→}{\mathrm{u}}}_{\mathrm{n}-1}\\ .& .& .\end{array}\right]$and A is A is an $\mathrm{n}×\mathrm{m}$matrix, a vector $\stackrel{→}{x}$in ${\mathrm{R}}^{\mathrm{n}}$called a least square solution of this system if $||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}{\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}||≤||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}}||$for all $\stackrel{→}{\mathrm{x}}$in ${\mathrm{R}}^{\mathrm{m}}$.

The term least square solution reflects the fact that minimizing the sum of the squares of the component of the vector $\left({\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}}\right)$.

To show ${\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}$of a linear system $\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$.

Consider the following string of equivalent statements.

The vector ${\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}$is a least square solution of the system $\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$.

Then by the definition as follows.

$⇔||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}{\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}||≤||{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}}||$for all$\stackrel{→}{\mathrm{x}}\mathrm{in}{\mathrm{R}}^{\mathrm{m}}$

Also by the theorem 5.4.3 as follows.

$⇔\mathrm{A}\stackrel{→}{\mathrm{x}}=\mathrm{proj}\left(\stackrel{→}{\mathrm{b}}\right)$where $\mathrm{V}=\mathrm{im}\left(\mathrm{A}\right)$.

Similarly by the theorem 5.1.4 and 5.4.1 as follows.

$⇔{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}}$is in ${\mathrm{V}}^{âŠ\yen }=(\mathrm{imA}{)}^{âŠ\yen }=\mathrm{keR}({\mathrm{A}}^{\mathrm{T}})$

$$\begin{gathered} ⇕ \\ {\mathrm{A}}^{\mathrm{T}}({\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}-\mathrm{A}\stackrel{→}{\mathrm{x}})=\stackrel{→}{0} \\ ⇕ \\ {\mathrm{A}}^{\mathrm{T}}\mathrm{A}{\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}={\mathrm{A}}^{\mathrm{T}}{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}} \end{gathered}$$

Therefore, the least square of the system$\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$ are the exact solution of the system ${\mathrm{A}}^{\mathrm{T}}\mathrm{A}{\stackrel{→}{\mathrm{x}}}^{\mathrm{k}}={\mathrm{A}}^{\mathrm{T}}{\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$.

Thus, the system is called the normal equation of $\mathrm{A}\stackrel{→}{\mathrm{x}}={\stackrel{→}{\mathrm{u}}}_{\mathrm{n}}$.