Prove eigenvalues are real

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In summary, eigenvalues are a mathematical concept used in linear algebra to describe the behavior of a linear transformation on a vector space. It is important to prove that eigenvalues are real because it allows for accurate predictions and interpretations in real-world scenarios. There are various methods to prove that eigenvalues are real, but there are exceptions to this rule, such as when the linear transformation is not defined on a real vector space or has repeated eigenvalues. Real eigenvalues are preferred over complex eigenvalues due to their practical applications and simpler interpretations.
  • #1
andrewm
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Given a 4x4 non-Hermitian matrix, is there any method I can use to prove the eigenvalues are real, aside from actually computing them?

I'm looking for something like the converse of the statement "M is Hermitian implies M has real eigenvalues".

When can one say that the eigenvalues of a given matrix are real?
 
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  • #2
In general, the eigenvalues of a non-Hermitian matrix can be complex. You would need to compute them.
 
  • #3
OK, I'm not surprised. Thanks anyways.
 

1. What are eigenvalues?

Eigenvalues are a mathematical concept used in linear algebra to describe the behavior of a linear transformation on a vector space. They represent the scalar factors by which a given vector is scaled when it is transformed by a linear transformation.

2. Why is it important to prove that eigenvalues are real?

Proving that eigenvalues are real is important because it allows us to make accurate predictions about the behavior of linear transformations. If eigenvalues were complex, the results of linear transformations would be more difficult to interpret and apply in real-world scenarios.

3. How can we prove that eigenvalues are real?

There are a few different ways to prove that eigenvalues are real. One common method is to use the algebraic definition of eigenvalues, which involves solving a characteristic polynomial. Another method is to use the geometric definition of eigenvalues, which involves analyzing the behavior of a linear transformation on a vector space.

4. Are there any exceptions to the rule that eigenvalues are real?

Yes, there are a few exceptions to the rule that eigenvalues are real. One exception is when the linear transformation is not defined on a real vector space, such as in the case of a complex vector space. Another exception is when the linear transformation has repeated eigenvalues, which can result in complex eigenvalues.

5. Why do we only care about real eigenvalues and not complex eigenvalues?

We primarily care about real eigenvalues because they are easier to work with and have more practical applications. In many cases, real eigenvalues represent physical quantities such as energy levels or frequencies in a system. Additionally, real eigenvalues often have simpler and more intuitive interpretations compared to complex eigenvalues.

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