91Ó°ÊÓ

Eigenvectors are a nonzero vector that is mapped by a given linear transformation of a vector space onto a vector that is the product of a scalar multiplied by the original vector.

The objective is to find a basis of the linear space V of all $3×3$matrices for which $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$and $\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$are eigenvectors, and determine the dimension of V.

Let $\stackrel{→}{v}$be an eigenvector of A then $A\stackrel{→}{v}=\mathrm{λ}\stackrel{→}{v}$.

Let $A=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]$.

As $\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$and $\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]$are eigenvectors of a $3×3$matrix A, then

$$\begin{gathered} \left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]={\mathrm{λ}}_1\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right] \\ \left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right]={\mathrm{λ}}_2\left[\begin{array}{c}0\\ 0\\ 1\end{array}\right] \end{gathered}$$

Solve the values of the entries in A.

$\left[\begin{array}{c}a\\ d\\ g\end{array}\right]=\left[\begin{array}{c}{\mathrm{λ}}_1\\ 0\\ 0\end{array}\right]\mathrm{and}\left[\begin{array}{c}c\\ f\\ i\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ {\mathrm{λ}}_2\end{array}\right]$

From the above vectors,$a={\mathrm{λ}}_1,d=0,g=0,c=0,f=0,i={\mathrm{λ}}_2$ .

And b,e,h are free variables can take any nonzero values.

Thus, all such matrices are in the form:

$$\begin{gathered} A=\left[\begin{array}{ccc}a& b& 0\\ 0& e& 0\\ 0& h& i\end{array}\right] \\ =\left\{\begin{array}{c}a\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+b\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+e\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]\\ +h\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right]+i\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]\end{array}\right\} \end{gathered}$$

Therefore, the dimension V of is dim V = 5 and its basis is:

$\left\{[\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 1& 0\end{array}\right]+\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 1\end{array}\right]]\right\}$