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Another eigenvalue problem.

  • Thread starter snoggerT
  • Start date
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Find a 2\times 2 matrix A for which

E_4 = span [1,-1] and E_2 = span [-5, 6]

where E_(lambda) is the eigenspace associated with the eigenvalue (lambda)


relevant equations: Av=(lambda)v

3. The Attempt at a Solution

I've pretty much gotten most of the eigenspace/value problems down, but this one I'm clueless on how to work. Seems that you would have to work backwards , but I don't know how to do that.
 

Answers and Replies

CompuChip
Science Advisor
Homework Helper
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Suppose you have a matrix A and it is diagonalizable. That means you can write it as
[tex]A = C D C^{-1}[/tex]
When A is given, how can you construct the matrices C and D?

Once you answer this question, you'll also see the answer to your problem.
 
HallsofIvy
Science Advisor
Homework Helper
41,738
897
You can just do it directly. You are told that
[tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}1 \\ -1\end{array}\right)= 4\left(\begin{array}{c}1\\ -1\end{array}\right)= \left(\begin{array}{c}4\\ -4\end{array}\right)[/tex]
which gives you two equations for a, b, c, d and
[tex]\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}-5 \\ 6\end{array}\right)= 2\left(\begin{array}{c}-5\\ 6\end{array}\right)= \left(\begin{array}{c}-10\\ 12\end{array}\right)[/tex]
which gives you another two equations. Solve those 4 equations.
 

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