Getting eigenvalues of an arbitrary matrix with programming

In summary, power iteration is a method for finding the dominant eigenvalue and corresponding eigenvector of a matrix A. It involves starting with a random compatible vector and iteratively multiplying it by A and normalizing it until it converges to the desired values. The inverse power iteration can also be used to find the least dominant eigenvalue. There are computer methods, such as scipy.linalg.eig, that can be used to find all eigenvalues of a matrix. The algorithm for this method is known as Rayleigh Quotient Iteration.
  • #1
Trollfaz
140
14
I have learnt about the power iteration for any matrix say A.
How it works is that we start with a random compatible vector v0. We define vn+1 as
vn+1=( Avn)/|max(Avn)|
After an arbitrary large number of iterations vn will slowly converge to the eigenvector associated with the dominant eigenvalue of A. To get the dominant eigenvalue, simply divide Avn by vn. The inverse power iteration can be used to find the least dominant eigenvalue of A by performing the iterations on A-1 assuming A is not singular else, the least dominant eigenvalue of A is automatically 0.
Heres how my function looks like on python:
def dominant_eigen(matrix,precision):
v=random_vector(len(matrix))
for i in range(precision):
v=matrix_vector_product(matrix,v) #takes in A and v and returns Av
v=tuple(map(lambda x:x/infinity_norm_v(v),v)) #infinity_norm_v returns magnitude of the dominant row entry
Av=matrix_vector_product(matrix,v)
return {'eigenvector':v,'eigenvalue':Av[0]/v[0]}
Are there computer methods to get all the eigenvalues of any matrices?
 
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  • #2

FAQ: Getting eigenvalues of an arbitrary matrix with programming

1. What is the importance of eigenvalues in programming?

Eigenvalues are important in programming because they provide valuable information about the behavior and properties of a matrix. They can be used to determine the stability of a system, identify patterns and relationships within data, and optimize algorithms.

2. How can I calculate eigenvalues of a matrix using programming?

There are various algorithms and methods for calculating eigenvalues of a matrix using programming, such as the power method, QR algorithm, and Jacobi method. These methods involve iterative processes and can be implemented in programming languages such as Python, MATLAB, and R.

3. Can I get complex eigenvalues using programming?

Yes, it is possible to get complex eigenvalues using programming. In fact, many real-world systems and data sets have complex eigenvalues, which can provide insights into the underlying dynamics and relationships.

4. Is there a limit to the size of matrix that can be used to calculate eigenvalues with programming?

The size of the matrix that can be used to calculate eigenvalues with programming depends on the computational resources available. However, there are methods for calculating eigenvalues of very large matrices, such as the Arnoldi algorithm and the Lanczos algorithm.

5. Can eigenvalues be negative?

Yes, eigenvalues can be negative. The sign of an eigenvalue is determined by the properties of the matrix, such as symmetry or skew-symmetry, and does not affect its importance or usefulness in programming.

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