Setting Free variables when finding eigenvectors

In summary, after setting up equations for eigenvectors and finding the eigenvalues, there was confusion about which variable to use as a free variable. It was realized that both a and b can be used, and it was also noted that the sign of the resulting vectors may be different. It is important to note that any non-zero scalar multiple of an eigenvector is also an eigenvector.
  • #1
Sunwoo Bae
61
4
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?
 
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  • #2
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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FAQ: Setting Free variables when finding eigenvectors

What are free variables when finding eigenvectors?

Free variables are variables that can take on any value when solving for eigenvectors. They are typically denoted by the letter t and represent the degree of freedom in the eigenvector solution.

How do I determine the number of free variables in an eigenvector?

The number of free variables in an eigenvector is equal to the dimension of the eigenspace associated with the eigenvalue. This can be found by solving the characteristic equation and finding the multiplicity of the eigenvalue.

Why are free variables important in finding eigenvectors?

Free variables allow for a general solution to the eigenvector problem, as they represent the different possible directions of the eigenvector. Without free variables, the eigenvector solution would be limited to a single direction.

How do I use free variables to find the eigenvector basis?

To find the eigenvector basis, we first solve for the eigenspace associated with each eigenvalue. Then, we use the free variables to create linearly independent eigenvectors that span the eigenspace. These eigenvectors form the eigenvector basis.

Can free variables be negative or complex numbers?

Yes, free variables can take on any value, including negative and complex numbers. This allows for a more general solution to the eigenvector problem and allows for complex eigenvalues and eigenvectors to be included in the solution.

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