Are Both Eigenvectors Correct?

In summary, corresponding eigenvectors are a set of vectors associated with the same eigenvalue in a linear transformation. They are found by solving an equation involving the matrix representing the transformation, the eigenvalue, and the eigenvector. These eigenvectors are significant in understanding the behavior of a transformation and are used in various real-world applications such as physics, engineering, and computer graphics. It is possible for a linear transformation to have multiple corresponding eigenvectors for the same eigenvalue, with the number of corresponding eigenvectors being equal to the multiplicity of the eigenvalue.
  • #1
Cpt Qwark
45
1
say for example when I calculate an eigenvector for a particular eigenvalue and get something like
[tex]\begin{bmatrix}
1\\
\frac{1}{3}
\end{bmatrix}[/tex]

but the answers on the book are

[tex]\begin{bmatrix}
3\\
1
\end{bmatrix}[/tex]

Would my answers still be considered correct?
 
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  • #2
Yes, multiplying an eigenvector with a scalar yields another eigenvector (makes kind of sense right? Think about the definition). Usually people will multiply the result with whatever scalar makes all the entries integers for representation purposes, but both results are correct.
 
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FAQ: Are Both Eigenvectors Correct?

What are corresponding eigenvectors?

Corresponding eigenvectors are a set of vectors that are associated with the same eigenvalue in a linear transformation. They represent the directions in which the transformation only scales the vector without changing its direction.

How do you find corresponding eigenvectors?

To find corresponding eigenvectors, you need to first find the eigenvalues of the linear transformation. Then, for each eigenvalue, you solve the equation (A-λI)x=0, where A is the matrix representing the transformation, λ is the eigenvalue, and x is the corresponding eigenvector.

What is the significance of corresponding eigenvectors?

Corresponding eigenvectors are important because they help us understand the behavior of a linear transformation. They also provide a way to simplify calculations and make predictions about the transformation.

Can corresponding eigenvectors be different for the same eigenvalue?

Yes, it is possible for a linear transformation to have multiple corresponding eigenvectors for the same eigenvalue. In fact, the number of corresponding eigenvectors for an eigenvalue is equal to the multiplicity of that eigenvalue.

How are corresponding eigenvectors used in real-world applications?

Corresponding eigenvectors have various applications in fields such as physics, engineering, and computer graphics. They are used to analyze and understand the behavior of systems, make predictions, and solve complex problems efficiently.

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