91影视

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

Given that, A is any symmetric 2x2 matrix with eigenvalues -2 and 3, and is a unit vector ${\mathrm{鈩�}}^2$. From the spectral theorem, we know that there exists an orthonormal eigen basis ${\stackrel{鈫�}{v}}_1,{\stackrel{鈫�}{v}}_2$for T , with associated real eigenvalues ${位}_1=3$and ${位}_2=-2$(Arrange things so that $\left|{位}_1\right|鈮�\left|{位}_2\right|$). Now consider the unit vector $\stackrel{鈫�}{u}$is represented below:

$$\begin{gathered} \stackrel{鈫�}{u}=c_1{\stackrel{鈫�}{v}}_1+c_2{\stackrel{鈫�}{v}}_2 \\ 鈬�A\stackrel{鈫�}{u}={位}_1{c}_1{\stackrel{鈫�}{v}}_1+{位}_2{c}_2{\stackrel{鈫�}{v}}_2 \\ 鈬�A\stackrel{鈫�}{u}=3c_1{\stackrel{鈫�}{v}}_1+2c_2{\stackrel{鈫�}{v}}_2 \end{gathered}$$

Now evaluate $\stackrel{鈫�}{u}.A\stackrel{鈫�}{u}$as follows:

role="math" localid="1659612634271" $$\begin{gathered} \stackrel{鈫�}{u}.A\stackrel{鈫�}{u}=\left(c_1{\stackrel{鈫�}{v}}_1+c_2{\stackrel{鈫�}{v}}_2\right).\left(3c_1{\stackrel{鈫�}{v}}_1-2c_2{\stackrel{鈫�}{v}}_2\right) \\ 鈬�\stackrel{鈫�}{u}.A\stackrel{鈫�}{u}=3c_1^2-2c_2^2\left(1\right) \end{gathered}$$

$3c_1^2-2c_2^2鈮�\left(3c_1^2+3c_2^2=3\right)\left(2\right)$

and

$\left(-2c_2^2-2c_2^2=-2\right)鈮�3c_1^2-2c_2^2\left(3\right)$

From (1), (2) and (3) we can imply that the possible values of the dot product$\stackrel{鈫�}{u}.A\stackrel{鈫�}{u}$is as represented below:

$-2鈮�\stackrel{鈫�}{u}.A\stackrel{鈫�}{u}鈮�3$

The orthonormal Eigen values are here ${位}_1=3$is positive and ${位}_2=-2$is negative. The plot of the unit circle and its image under A is represented below.