You can get a sense for the timings involved for different sizes: In : M = make_matrix(5) Unfortunately exact rational numbers do not have O(1) arithmetic cost so in fact the complexity is significantly worse than O(N**4) as you scale up to larger matrices. Finding the characteristic polynomial this way is roughly O(N**4) for an N x N dense matrix if the underlying coefficient field has O(1) arithmetic cost. This is much more efficient than using determinants to find the characteristic polynomial but still slow for large matrices. This uses the Berkowicz algorithm to compute the characteristic polynomial which is the slowest part of the operation (the charpoly method). """Random matrix of rational numbers with all integer eigenvalues""" This is usually only worth doing if your original matrix has exact rational numbers (not floats) but it should obtain precisely the real eigenvalues: from sympy import * In any case here is some code that can show how to get the exact real eigenvalues of a matrix more efficiently using SymPy. If your initial matrix is a matrix of floats though then it is already approximate in which case there is a good chance that there will be no benefit in using SymPy. Exact or symbolic calculations are a lot slower than fixed precision floating point though so you should expect things to slow down a lot compared to using NumPy or SciPy. Determining that the eigenvalues are precisely real will require exact arithmetic in general. The advantage of using SymPy rather than NumPy or SciPy in this context is just if you want to perform the calculation exactly or symbolically. I can throw more hardware at it on a cluster, but have not been able to try this yet. However this does not output an answer in 12 hours with any amount of hardware thrown at it. T = np.loadtxt('rep10_T_ij.dat', delimiter=' ')ĭisplay(roots(poly(char_poly, domain=CC))) My understanding is that this will force answers as real numbers. ![]() Is it possible somehow to find complex eigenvalues using SymPy? Using sympy as described here makes sense: I turned to the sympy library, which also returned complex numbers as the solution. The scipy and numpy libraries return complex numbers - which is apparently due to not being able to solve the characteristic polynomial as real numbers or the algorithms being optimised to do this. Columns do not have to sum to anything in particular. Each row of the matrix, T, sums to 1.0 (100% probability). Elements of the matrix, T, are probabilities from 0 to 1. This is demonstrating in the MATLAB code below.Am trying to determine the (real) left-eigenvectors and eigenvalues of a 100 by 100 matrix. Going through the same process for the second eigenvalue:Īgain, the choice of the +1 and -2 for the eigenvectors was arbitrary only their ratio is essential. If we didn't have to use +1 and -1, we have used any two quantities of equal magnitude and opposite sign. In this case, we find that the first eigenvector is any 2 component column vector in which the two items have equal magnitude and opposite sign. Let's find the eigenvector, v 1, connected with the eigenvalue, λ 1=-1, first. ![]() Example: Find Eigenvalues and Eigenvectors of the 2x2 MatrixĪll that's left is to find two eigenvectors. For each eigenvalue, there will be eigenvectors for which the eigenvalue equations are true. We will only handle the case of n distinct roots through which they may be repeated. These roots are called the eigenvalue of A. This equation is called the characteristic equations of A, and is a n th order polynomial in λ with n roots. If vis a non-zero, this equation will only have the solutions if The eigenvalues problem can be written as The vector, v, which corresponds to this equation, is called eigenvectors. It is also called the characteristic value. ![]() ![]() Any value of the λ for which this equation has a solution known as eigenvalues of the matrix A. In this equation, A is a n-by-n matrix, v is non-zero n-by-1 vector, and λ is the scalar (which might be either real or complex). Next → ← prev Eigenvalues and EigenvectorsĪn eigenvalues and eigenvectors of the square matrix A are a scalar λ and a nonzero vector v that satisfy
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