How Can You Create a Math Problem Involving Eigenvalues and Matrix Powers?

[nonequilibrium]

Hi,

I'd like to make my own mathematical problem in which you'd have to use the calculation of eigenvalues, eigenvectors and a (high) power of a matrix A. (with definitions: AX = [tex]\lambda[/tex]X & A[tex]^{n}[/tex] = BD[tex]^{n}[/tex]B[tex]^{-1}[/tex]).

I'm not searching for something too complicated, it's more like to integrate these methods into your mathematical skills.

All my tries have failed so far. As I want to take the power of a matrix, I'm thinking of Markov chains or Lesliematrices. Does anybody have some suggestions?

I'd also prefer there not being a need of ICT-use.

 

[nate808]

Hello,

Thank you for your post. I understand your desire to integrate the concepts of eigenvalues, eigenvectors, and matrix powers into a mathematical problem. Here is one possible problem that may fulfill your requirements:

Consider a population of rabbits on an island, where the population is modeled by a Leslie matrix. The Leslie matrix is a square matrix that represents the reproductive rates and survival rates of different age groups in the population. The matrix is defined as follows:

L = [f_1, f_2, f_3, ..., f_n
s_1, 0, 0, ..., 0
0, s_2, 0, ..., 0
0, 0, s_3, ..., 0
..., ..., ..., ..., ...
0, 0, 0, ..., s_n ]

Where f_i represents the fertility rate of individuals in age group i, and s_i represents the survival rate of individuals in age group i.

Now, let's define a new matrix A, where A = L - I, and I is the identity matrix. This means that each element in the matrix is subtracted by 1. This new matrix represents the change in population from one time step to the next.

To find the eigenvalues and eigenvectors of A, we can use the standard methods of finding eigenvalues and eigenvectors for matrices. However, instead of raising A to a high power, we can use the definition you provided in your post: A^n = BD^nB^-1. Here, D is a diagonal matrix with the eigenvalues of A on the diagonal, and B is a matrix with the corresponding eigenvectors of A as its columns.

Now, let's say we want to predict the population of rabbits after 10 time steps. We can use the matrix power A^10 to do this. However, instead of using traditional matrix multiplication, we can use the definition A^n = BD^nB^-1. This means that we first find the eigenvalues and eigenvectors of A, then raise the eigenvalues to the 10th power, and then multiply by the eigenvector matrix and its inverse.

The resulting matrix will give us the predicted population of rabbits after 10 time steps. We can then compare this with the actual population after 10 time steps to see how accurate our prediction was.

I hope this problem meets your requirements and provides a useful application of eigenvalues, eigenvectors

 

[Architeuthis Dux]

I would suggest creating a problem using the concept of diagonalization. This involves finding the eigenvalues and eigenvectors of a matrix, and then using them to transform the original matrix into a diagonal matrix. This process can be applied to any square matrix, and it is a fundamental concept in linear algebra.

Here's an example problem:

Consider the following matrix A:
A = [1 2
2 3]

1. Find the eigenvalues and eigenvectors of A.
2. Use the eigenvalues and eigenvectors to diagonalize A, i.e. find a diagonal matrix D and an invertible matrix B such that A = BDB^-1
3. Find A^10 using the diagonalized form.
4. Use the diagonalized form to find A^n for any positive integer n.
5. Compare the results obtained in steps 3 and 4 with the direct calculation of A^10 and A^n using the definition A^n = BD^nB^-1.

This problem integrates the concepts of eigenvalues, eigenvectors, and powers of a matrix, and it can be solved without the use of ICT. You can also modify the problem by changing the values in matrix A or by using a different matrix altogether. This will allow for a variety of solutions and further exploration of the concept.