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Homework Statement
Given the linear transformation l : R 2 → R 2 defined below, find characteristic equation, real eigenvalues and corresponding eigenvectors. a) l(x, y) = (x + 5y, 2x + 4y)
Homework Equations
characteristic equation = det (A-λI) = 0
The Attempt at a Solution
l(x, y) = (x + 5y, 2x + 4y)
A =
[ 1 + 5 ]
[ 2 + 4 ]
det (A-λI) = 0 =
[ 1-λ + 5 ]
[ 2 + 4-λ ]
determinant works out to be
D =
(λ2 - 5λ - 6) = 0
so eigenvalues are -1 and 6.
D =
[-1 0]
[ 0 6]
to get eigenvectors we multiply the original matrix A by D
[1 5 ] [ -1 0]
[ 2 4] [ 0 -1]
=
[ -1 -5 ]
[ -2 -4 ]
This is where I'm stuck
=
[ -1 -5 ] * [x] = 0
[ -2 -4 ] [y]
-x-5y = 0
-2x-4y = 0
I can't see how x in the first equation can = x in the second equation. The same with both y's. How can I find the eigenvector here? I'm going back through my steps now to see if I made an error somewhere. I'm having the same trouble finding eigenvectors with the second eigenvalue of 6 as well.
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