91Ó°ÊÓ

Take A's be${\mathrm{λ}}_1,{\mathrm{λ}}_2,{\mathrm{λ}}_3,......,{\mathrm{λ}}_n$ eigenvalues as an example. Let${\mathrm{λ}}_1â‰\yen {\mathrm{λ}}_2â‰\yen ...â‰\yen {\mathrm{λ}}_n$ be the set of all unit vectors in$\mathrm{q}\left({\mathrm{S}}^{\mathrm{n}-1}\right)=[{\mathrm{λ}}_{\mathrm{n}},{\mathrm{λ}}_1]$ , where${\mathrm{S}}^{\mathrm{n}-1}$ represents the set of all unit vectors in without losing generality. That is to say,

$$\begin{gathered} {\mathrm{λ}}_1â‰\yen \mathrm{q}\left(\mathrm{x}\right)â‰\yen {\mathrm{λ}}_{\mathrm{n}}\mathrm{for}\mathrm{all}\mathrm{x}∈{\mathrm{S}}^{\mathrm{n}-1} \\ {\mathrm{λ}}_1â‰\yen {\mathrm{x}}^{\mathrm{T}}\mathrm{Ax}â‰\yen {\mathrm{λ}}_{\mathrm{n}}\mathrm{for}\mathrm{all}\mathrm{x}∈{\mathrm{S}}^{\mathrm{n}-1} \end{gathered}$$

Consider the vector$e_1=\left(1,0,....,0\right)∈{\mathrm{R}}^{\mathrm{n}}$, then $||e_1||$= 1and thus,${\mathrm{e}}_1∈{\mathrm{S}}^{\mathrm{n}-1}$ . Thus, from the above discussion, we get

${\mathrm{λ}}_1â‰\yen {\mathrm{e}}^{\mathrm{T}}{\mathrm{Ae}}_1={\mathrm{a}}_{11}â‰\yen {\mathrm{λ}}_{\mathrm{n}}$

As a result, ${\mathrm{λ}}_1â‰\yen {\mathrm{a}}_{11}$. As a result, for a symmetric matrix ,A an eigenvalue of A exists, namely A, viz, ${\mathrm{λ}}_1$the greatest eigenvalue of such that ${\mathrm{λ}}_1â‰\yen {\mathrm{a}}_{11}$.

$\mathrm{q}\left({\mathrm{S}}^{\mathrm{n}-1}\right)=[{\mathrm{λ}}_{\mathrm{n}},{\mathrm{λ}}_1]$where ${\mathrm{S}}^{\mathrm{n}-1}$refers to the set of unit vectors in${\mathrm{R}}^{\mathrm{n}}$ , and${\mathrm{λ}}_{\mathrm{n}}$ and${\mathrm{λ}}_1$ are the smallest and greatest eigenvalues of A, respectively (Exercise 27).