Solving Complex Eigenvalues Homework

In summary: They are a complex conjugate pair.In summary, the conversation discusses the application of the eigenvalue method to find a general solution for a given system of equations. The attempt at a solution involves finding the eigenvalues and eigenvectors of the system, but there is an error in finding the eigenvalues. The correct solution involves a complex conjugate pair of eigenvalues, resulting in a solution with complex coefficients.
  • #1
Totalderiv
70
1

Homework Statement


Apply the eigenvalue method to find a general solution of the given system.
[tex]x_1' = 5x_1 - 9x_2[/tex]
[tex]x_2' = 2x_1 - x_2[/tex]


Homework Equations


(A-λI)v=0


The Attempt at a Solution


[tex]x_1' = 5x_1 - 9x_2[/tex]
[tex]x_2' = 2x_1 - x_2[/tex]

[tex]\left[
\begin{array}{cc}
5-λ & -9\\
2 & -1-λ
\end{array}
\right]=(5-λ)(-1-λ)+18=0[/tex]
[tex]λ^2-4λ-13=0[/tex]
[tex](λ-2)^2 -9=0[/tex]
[tex]λ=2+3i,\overline{λ}=2-3i[/tex]
So I plugged λ into the matrix;
[tex]\left[
\begin{array}{cc}
3-3i & -9\\
2 & -3-3i
\end{array}
\right]
\left[
\begin{array}{cc}
a\\
b
\end{array}
\right]=(3-3i)a-9b=0
2a-(3+3i)b=0[/tex]
This is where I'm stuck...the answer is;
[tex]x_1(t)=3e^{2t}(c_1cos2t - 5c_2sin2t)[/tex]
[tex]x_2(t)=e^{2t}[(c_1+c_2)cos3t + (c_1-c_2)sin3t)][/tex]
 
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  • #2
Any one?
 
  • #3
Totalderiv said:

Homework Statement


Apply the eigenvalue method to find a general solution of the given system.
[tex]x_1' = 5x_1 - 9x_2[/tex]
[tex]x_2' = 2x_1 - x_2[/tex]


Homework Equations


(A-λI)v=0


The Attempt at a Solution


[tex]x_1' = 5x_1 - 9x_2[/tex]
[tex]x_2' = 2x_1 - x_2[/tex]

[tex]\left[
\begin{array}{cc}
5-λ & -9\\
2 & -1-λ
\end{array}
\right]=(5-λ)(-1-λ)+18=0[/tex]
[tex]λ^2-4λ-13=0[/tex]
[tex](λ-2)^2 -9=0[/tex]
[tex]λ=2+3i,\overline{λ}=2-3i[/tex]
This is your error. [itex](\lambda- 2)^2- 9= 0[/itex] is equivalent to [itex](\lambda- 2)^2= 9[/itex] so [itex]\lambda- 2= \pm 3[/itex]. There is no "i".

So I plugged λ into the matrix;
[tex]\left[
\begin{array}{cc}
3-3i & -9\\
2 & -3-3i
\end{array}
\right]
\left[
\begin{array}{cc}
a\\
b
\end{array}
\right]=(3-3i)a-9b=0
2a-(3+3i)b=0[/tex]
This is where I'm stuck...the answer is;
[tex]x_1(t)=3e^{2t}(c_1cos2t - 5c_2sin2t)[/tex]
[tex]x_2(t)=e^{2t}[(c_1+c_2)cos3t + (c_1-c_2)sin3t)][/tex]
There are no eigenvectors because 2+ 3i and 2- 3i are not eigenvalues.
 

Related to Solving Complex Eigenvalues Homework

1. What are complex eigenvalues and why are they important in solving homework problems?

Complex eigenvalues are values that satisfy a specific equation known as the characteristic equation. They are important in solving homework problems because they represent the solutions to systems of linear equations, which are often used in various scientific and mathematical applications.

2. How do I find the complex eigenvalues of a matrix?

To find the complex eigenvalues of a matrix, you first need to calculate the determinant of the matrix and then solve the characteristic equation using the quadratic formula. The solutions to the characteristic equation will be the complex eigenvalues of the matrix.

3. Can complex eigenvalues have real parts equal to zero?

Yes, complex eigenvalues can have real parts equal to zero. In fact, if a matrix has real coefficients, then its complex eigenvalues will always have real parts equal to zero.

4. How do I know if a matrix has complex eigenvalues?

A matrix has complex eigenvalues if its characteristic equation has complex solutions. This can be determined by calculating the discriminant of the characteristic equation, which will be negative if the solutions are complex.

5. Are there any special methods or techniques for solving problems involving complex eigenvalues?

Yes, there are special methods and techniques for solving problems involving complex eigenvalues, such as the power method, the QR algorithm, and the Jacobi method. These methods can be more efficient and accurate than using the standard quadratic formula for solving the characteristic equation.

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