Setting Free variables when finding eigenvectors

  • #1
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1

Homework Statement:

Confusion in finding eigenvector? (example shown below)

Relevant Equations:

matrix multiplication
upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations.

So I took b as a free variable to solve the equation int he following way.
1597304906485.png


But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free variable too.
1597305037416.png


But now I am confused because this results in vectors that is different in sign. Can anyone explain whether I should use a or b as a free variable?
 

Answers and Replies

  • #2
PeroK
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If ##\vec u## is an eigenvector (corresponding to a certain eigenvalue), then ##-\vec u## is also an eigenvector (corresponding to the same eigenvalue). Both your answers are correct.

In general, ##c \vec u## is also an eigenvector for any number ##c \ne 0##. Often you choose ##c## such that the eigenvector is normalised - i.e. has length ##1##,
 
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