Matrix Differential Equation with Generalized Eigenvectors

In summary, in this conversation, the speaker is asking for quick help with a differential equation before an exam. They share the equation and initial conditions and mention finding two eigenvalues and their corresponding eigenvector. They then discuss finding a generalized eigenvector and needing help with the form of the solution.
  • #1
patchwerk
3
0
Hey guys, need some quick help before an exam

I have a differential eqn.

x' = | 0 1 | *x , and initial conditions x(0) = |2|
| -25 10 | |3|

I find that there are two eigenvalues 5, and 5

The corresponding eigenvector to 5 is [1 5] (vertical)

So i need to find a generalized eigenvector,

I do so in the form

(A - 5lambda)v_2 = v_1

I then find that v_2 = [0 1 ] (vertically)

I don't know what form my solution should now be in

Please help, I have an exam at 7,

Thanks,

Evan
 
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  • #2
the matrix is R1: 0 1 and R2: -25 10
and the IC is x(0) = R1:2 R2:13

sorry, not sure hwo to enter matrices
 
  • #3
patchwerk said:
the matrix is R1: 0 1 and R2: -25 10
and the IC is x(0) = R1:2 R2:13

sorry, not sure hwo to enter matrices

R2: is 3 sorry!
 

Related to Matrix Differential Equation with Generalized Eigenvectors

1. What is a matrix differential equation with generalized eigenvectors?

A matrix differential equation with generalized eigenvectors is a type of differential equation where the unknown function is a matrix and the dependent variable is a vector. The matrix is multiplied by the derivative of the vector, and the resulting equation is solved for the vector.

2. How is a matrix differential equation with generalized eigenvectors different from a regular differential equation?

A regular differential equation involves only scalar variables, while a matrix differential equation with generalized eigenvectors involves both matrices and vectors. Additionally, in a regular differential equation, the solution is a single function, while in a matrix differential equation, the solution is a matrix of functions.

3. What are the applications of matrix differential equations with generalized eigenvectors?

Matrix differential equations with generalized eigenvectors have various applications in physics, engineering, and economics. They can be used to model dynamic systems, such as population growth or chemical reactions, and to analyze the stability of these systems.

4. How are generalized eigenvectors used in solving matrix differential equations?

Generalized eigenvectors are used in solving matrix differential equations because they provide a complete basis for the solution space of the matrix. This means that any solution to the differential equation can be expressed as a linear combination of generalized eigenvectors.

5. Are there any limitations or challenges in solving matrix differential equations with generalized eigenvectors?

One limitation of using generalized eigenvectors in solving matrix differential equations is that the matrix must have distinct eigenvalues. If the matrix has repeated eigenvalues, more advanced techniques such as Jordan canonical form may be needed. Additionally, the process of finding generalized eigenvectors can be computationally intensive, making it challenging for large matrices.

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