Diff EQ Repeated Complex Eigenvalues?

In summary, the conversation discusses the dimensions of a matrix that will result in repeated complex eigenvalues. An example is given to illustrate this concept, and it is noted that no equations are needed. The conversation then delves into the characteristic equation of a 2x2 matrix and how to determine if the roots will be imaginary.
  • #1
th3ownly
2
0
1. What dimensions of a matrix will give repeated complex Eigenvalues? Give an example
of one and show that it has repeated complex Eigenvalues.


2. No really equations needed?


The Attempt at a Solution



My attempt is a 2x2 which i don't think is right but here it is.

If the matrix were
we will use x as lambda

(x)' = [2+3i - λ 0 ] (x)
(y) ' = [ 0 2+3i-λ] (y)
This would yield the eigenvalues 2+3i, 2+3i


I just don't think my solution is right
 
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  • #2
If you mean the matrix [[2+3i,0],[0,2+3i]] has a double complex eigenvalue of 2+3i, I don't see what could be wrong with that.
 
  • #3
Why do you think it doesn't? In order for a non-trivial solution det(A-λI) = 0 right?

Do it out -- does that equation ever equal zero?

However I think there is a better way to go about doing this you've simply listed a property of a 2x2 matrix where the eigenvalues will be the diagonal elements as long as everything else is 0.

The characteristic equation of a 2x2 matrix:

[A-λ, B]
[C, D-λ]

Is a quadratic, right?

How do we know if the roots will be imaginary from looking at the characteristic equation?
 
  • #4
Feldoh said:
Why do you think it doesn't? In order for a non-trivial solution det(A-λI) = 0 right?

Do it out -- does that equation ever equal zero?

However I think there is a better way to go about doing this you've simply listed a property of a 2x2 matrix where the eigenvalues will be the diagonal elements as long as everything else is 0.

The characteristic equation of a 2x2 matrix:

[A-λ, B]
[C, D-λ]

Is a quadratic, right?

How do we know if the roots will be imaginary from looking at the characteristic equation?

necause there is no sign change?
 

1. What are repeated complex eigenvalues in differential equations?

Repeated complex eigenvalues in differential equations occur when the characteristic equation of the system has complex roots that are repeated. This means that the eigenvalues have the form a ± bi, where a and b are real numbers and i is the imaginary unit.

2. How do repeated complex eigenvalues affect the solution of a differential equation?

Repeated complex eigenvalues can lead to multiple solutions for a given initial value problem. The general solution of the differential equation will involve a linear combination of two or more linearly independent solutions, depending on the multiplicity of the eigenvalues.

3. How do we find the general solution for a differential equation with repeated complex eigenvalues?

To find the general solution for a differential equation with repeated complex eigenvalues, we first solve for the two linearly independent solutions using the standard procedure for complex eigenvalues. Then, we use the principle of superposition to combine these solutions and find the general solution.

4. Are there any special cases when dealing with repeated complex eigenvalues in differential equations?

Yes, there are two special cases when dealing with repeated complex eigenvalues. The first case is when the repeated eigenvalues have a multiplicity of two, in which case the two linearly independent solutions will have the form e^atcos(bt) and e^atsin(bt). The second case is when the repeated eigenvalues have a higher multiplicity, in which case the solutions will involve higher powers of t.

5. Can repeated complex eigenvalues lead to instability in a differential equation system?

Yes, repeated complex eigenvalues can lead to instability in a differential equation system. This can occur when the real part of the eigenvalue is positive, which means that the solutions will grow exponentially with time. In this case, the system is considered unstable and may not have a physically meaningful solution.

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