(adsbygoogle = window.adsbygoogle || []).push({}); 1. Find the eigenvalues and the eigenvectors corresponding to eigenvalues of the matrix

A = [tex]\left[\begin{array}{ccccc} 1 & 3 \\ 4 & 2 \end{array}\right][/tex]

3. The attempt at a solution

[tex](\lambda I - A)[/tex] = [tex]\lambda \left[\begin{array}{ccccc} 1 & 0 \\ 0 & 1 \end{array}\right] -[/tex] [tex]\left[\begin{array}{ccccc} 1 & 3 \\ 4 & 2 \end{array}\right][/tex]

[tex]\left[\begin{array}{ccccc} \lambda - 1 & -3 \\ -4 & \lambda - 2 \end{array}\right][/tex] [tex]\left(\begin{array}{ccc}x\\y\end{ar ray}\right) =[/tex] [tex]\left(\begin{array}{ccc}0\\0\end{ar ray}\right)[/tex]

det(λI- A) = 0

=> (λ-1)(λ-2)-12 = 0

λ^{2}-3λ-10=0

(λ+2)(λ-5) = 0

λ = -2, 5

My problem is how to find the eigenspaces corresponding to these eigenvalues.

We have two cases, the first one is when [tex]\lambda = 5[/tex]. In this case we have the following:

[tex]\left[\begin{array}{ccccc} 4 & -3 \\ -4 & 3 \end{array}\right][/tex] [tex]\left(\begin{array}{ccc}x\\y\end{ar ray}\right) =[/tex] [tex]\left(\begin{array}{ccc}0\\0\end{ar ray}\right)[/tex]

So to find the eigenvectors corresponding to [tex]\lambda = 5[/tex], I think I should solve the system

[tex]\left[\begin{array}{ccccc} 4 & -3 \\ -4 & 3 \end{array}\right][/tex]

4x-3y = 0.....(1)

-4x+3y = 0.....(2)

How can I solve this? I'm not sure how this is done (if I minus (1) from (2) to eliminate x then the y would be eliminated as well).

Thanks.

Roam

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# Homework Help: Finding Eigenvectors

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