91影视

In linear algebra, a symmetric matrix is a square matrix that stays unchanged when its transpose is calculated. That instance, a symmetric matrix is one whose transpose equals the matrix itself.

Given that, A is any symmetric matrix $2脳2$with eigenvalues -2 and 3 , and $\stackrel{鈫�}{u}$is a unit vector in ${\mathrm{鈩�}}^2,{\mathrm{鈩�}}^2$. From the spectral theorem we know that there exists an orthonomal eigenbasis ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2$for T , with associated real eigenvalues ${位}_1=3$and ${位}_2=-2$(Arrange things so that $\left|{位}_1\right|鈮�\left|{位}_2\right|$). The unit circle in ${\mathrm{鈩�}}^2$comprises of all vectors of the form:

$\overline{)\stackrel{鈫�}{v}=\cos \left(t\right){\stackrel{鈫�}{v}}_1+\sin \left(t\right){\stackrel{鈫�}{v}}_2}$

The image of the unit circle comprises of the vectors

$$\begin{gathered} T\stackrel{鈫�}{v}=\cos \left(t\right)T\left({\stackrel{鈫�}{v}}_1\right)+\sin \left(t\right)T\left({\stackrel{鈫�}{v}}_2\right) \\ =\cos \left(t\right){位}_1{\stackrel{鈫�}{v}}_1+\sin \left(t\right){位}_2{\stackrel{鈫�}{v}}_2 \\ 鈬�T\left(\stackrel{鈫�}{v}\right)=\cos \left(t\right)\left(3\right){\stackrel{鈫�}{v}}_1+\sin \left(t\right)\left(-2\right){\stackrel{鈫�}{v}}_2 \\ \left({位}_1=3and{位}_2=-2\right) \end{gathered}$$

an ellipse whose semi major axis $\left(3\right){\stackrel{鈫�}{v}}_1$has the length

$||\left(3\right){\stackrel{鈫�}{v}}_1||=\left|\left(3\right)\right|=3,$

while the length of the semi minor axis is

$||\left(-2\right){\stackrel{鈫�}{v}}_2||=\left|\left(-2\right)\right|=2,$

Therefore,

$$\begin{gathered} \left|lowesteigenvalueofA\left|鈮�||A\stackrel{鈫�}{u}||鈮�\right|highesteigenvalueofA\right| \\ 鈬�\left|-2\right|鈮�||A\stackrel{鈫�}{u}||鈮�\left|3\right| \\ 鈬�-2鈮�||A\stackrel{鈫�}{u}||鈮�3 \end{gathered}$$

($\stackrel{鈫�}{u}$ is a unit vector)