Find a 2x2 Matrix A for Given Eigenspaces E_2 and E_4

[snoggerT]

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

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.

 

[CompuChip]

Suppose you have a matrix A and it is diagonalizable. That means you can write it as
$$A = C D C^{-1}$$
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]

You can just do it directly. You are told that
$$\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}1 \\ -1\end{array}\right)= 4\left(\begin{array}{c}1\\ -1\end{array}\right)= \left(\begin{array}{c}4\\ -4\end{array}\right)$$
which gives you two equations for a, b, c, d and
$$\left(\begin{array}{cc}a & b \\ c & d\end{array}\right)\left(\begin{array}{c}-5 \\ 6\end{array}\right)= 2\left(\begin{array}{c}-5\\ 6\end{array}\right)= \left(\begin{array}{c}-10\\ 12\end{array}\right)$$
which gives you another two equations. Solve those 4 equations.