91Ó°ÊÓ

• A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
• For example, is a two-row, three-column matrix

Consider an orthogonal matrix R whose first column is $\stackrel{â‡\P Ä}{\mathrm{v}}$which form the symmetric matrix $A=\stackrel{â‡\P Ä}{\mathrm{v}}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}$.

Since R is orthogonal it can only have Eigen values -1 and 1 .

Therefore, will have entries 0,-1 , and 1.

Now evaluate localid="1659606220772" $\mathrm{A}\stackrel{â‡\P Ä}{\mathrm{v}}$as follows:

$$\begin{gathered} \stackrel{â‡\P Ä}{\mathrm{v}}=\left(\stackrel{â‡\P Ä}{\mathrm{v}}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}\right)(\mathrm{Since}\mathrm{A}=\stackrel{â‡\P Ä}{\mathrm{v}}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}) \\ \stackrel{â‡\P Ä}{\mathrm{v}}=\stackrel{â‡\P Ä}{\mathrm{v}}\left({\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}\stackrel{â‡\P Ä}{\mathrm{v}}\right) \\ =\stackrel{â‡\P Ä}{\mathrm{v}}\left(1\right) \\ ⇒\mathrm{A}\stackrel{â‡\P Ä}{\mathrm{v}}=\stackrel{â‡\P Ä}{\mathrm{v}} \end{gathered}$$

(Since ${\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}\stackrel{â‡\P Ä}{\mathrm{v}}=1$)

Now evaluate $\mathrm{A}{\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}\left(\mathrm{i}=1,2,3,...,\mathrm{n}\right)$as follows:

$$\begin{gathered} \mathrm{A}{\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}=\left(\stackrel{â‡\P Ä}{\mathrm{v}}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}\right){\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}\left(\mathrm{Since}\right)\mathrm{A}=\stackrel{â‡\P Ä}{\mathrm{v}}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}} \\ =\stackrel{â‡\P Ä}{\mathrm{v}}\left({\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}{\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}\right) \\ =\stackrel{â‡\P Ä}{\mathrm{v}}\left(0\right) \\ ⇒\mathrm{A}{\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}=0\left(\mathrm{Since}{\stackrel{â‡\P Ä}{\mathrm{v}}}^{\mathrm{T}}{\stackrel{â‡\P Ä}{\mathrm{v}}}_{\mathrm{i}}=0\right) \end{gathered}$$

Hence, we get the orthogonal matrix S as:

S = R

And the diagonal matrix D that satisfies ${\mathrm{S}}^{-1}\mathrm{AS}=\mathrm{D}$as:

$$\begin{gathered} \mathrm{D}=\left[\begin{array}{cccccc}1& 0& 0& ...& 0& 0\\ 0& 0& 0& …& 0& 0\\ 0& 0& 0& …& 0& 0\\ â‹\circledR & â‹\circledR & â‹\circledR & â‹\pm & â‹\circledR & â‹\circledR \\ 0& 0& 0& …& 0& 0\\ 0& 0& 0& …& 0& 0\end{array}\right] \\ \mathrm{Therefore},\mathrm{the}\mathrm{diagonal}\mathrm{matrix}\mathrm{is}\mathrm{S}=\mathrm{R} \\ \mathrm{D}=\left[\begin{array}{cccccc}1& 0& 0& ...& 0& 0\\ 0& 0& 0& …& 0& 0\\ 0& 0& 0& …& 0& 0\\ â‹\circledR & â‹\circledR & â‹\circledR & â‹\pm & â‹\circledR & â‹\circledR \\ 0& 0& 0& …& 0& 0\\ 0& 0& 0& …& 0& 0\end{array}\right] \end{gathered}$$