Non-homogeneous systems with repeated eigenvalues

In summary, you can solve non-homogeneous systems with repeated eigenvalues using the same methods as for non-repeated eigenvalues, such as variation of parameters and undetermined coefficients. The fundamental matrix will contain the solution with the generalized eigenvalue. This is because the fundamental matrix is defined by solving a Cauchy problem and can be used to find the solution to the initial inhomogeneous system.
  • #1
faradayscat
57
8
Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways? i.e. variation of parameters, undetermined coefficients, etc... would the fundamental matrix contain the solution with the generalized eigenvalue?

Thanks!
 
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  • #2
faradayscat said:
Quick question, can you solve non-homogeneous systems with repeated eigenvalues the same ways?
Sure you can. Actually this question is not about eigenvalues

Consider a system ##\dot x=A(t)x+b(t),\quad x\in\mathbb{R}^m##, here ##A(t)## is a square matrix with coefficients depending on t.
The fundamental matrix ##X(t)## is defined by means of the Cauchy problem ##\dot X(t)=A(t)X,\quad X(0)=I##. Then the solution ##x(t),\quad x(0)=\hat x## to the initial inhomogeneous system is given by the formula
##x(t)=X(t)\hat x+\int_0^t X(t)X^{-1}(s)b(s)ds##
 

1. What are non-homogeneous systems with repeated eigenvalues?

Non-homogeneous systems with repeated eigenvalues are systems of linear equations where the matrix has at least one eigenvalue with a multiplicity greater than one. This means that the eigenvalue appears multiple times in the characteristic equation, resulting in repeated roots.

2. How do you solve non-homogeneous systems with repeated eigenvalues?

To solve non-homogeneous systems with repeated eigenvalues, you first need to find the generalized eigenvectors associated with each repeated eigenvalue. Then, you can use these vectors to construct the Jordan canonical form of the matrix, which can be used to solve the system.

3. Why do non-homogeneous systems with repeated eigenvalues require a different approach to solve?

Non-homogeneous systems with repeated eigenvalues require a different approach to solve because repeated eigenvalues result in a loss of linear independence, making the standard methods of solving linear systems ineffective. Additionally, the lack of linear independence means there may be multiple solutions to the system.

4. Can non-homogeneous systems with repeated eigenvalues have a unique solution?

No, non-homogeneous systems with repeated eigenvalues typically do not have a unique solution. This is because the lack of linear independence in the system results in multiple solutions. However, in some cases, a unique solution may be possible if additional constraints or information are provided.

5. How are non-homogeneous systems with repeated eigenvalues used in real-world applications?

Non-homogeneous systems with repeated eigenvalues are commonly used in physics and engineering to model systems with periodic or oscillatory behavior. They are also used in statistics and data analysis to identify patterns and trends in large datasets. Additionally, they are used in machine learning algorithms for tasks such as image and speech recognition.

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