Computing eigenvectors and eigenvalues

Question

I have a (non-sparse) $9 \times 9$ matrix and I wish to obtain its eigenvalues and eigenvectors. Of course, the eigenvalues can be quite a pain as we will probably not be able to find the zeros of its characteristic polynomial.

Actually, what I really want is find the eigenvector belonging to the largest eigenvalue. Would the following code give me this: Eigenvectors[matrix, 1]?

How does Mathematica do this? Is there some kind of algorithm of which's existence I am unaware that computes the eigenvector belonging to the largest eigenvalue?

Can the largest eigenvalue be computed numerically (in modulus or even throw away all the complex ones), and then the eigenvectors pseudo-analytically using Eigenvectors?

Added: The matrix is symbolical.

Answer 1

The function to obtain both the eigenvalues and the eigenvectors is Eigensystem. Use it as {eigVals,eigVecs} = Eigensystem[matrix].

If the matrix is symbolic, then the output (if you wait long enough for it to churn out an answer!) will only be as a list of general solutions for the roots of a 9th order polynomial with unknown coefficients, and there are no closed form solutions for polynomials with orders greater than 4. The results will not have any particular ordering.

On the other hand, a 9x9 numerical matrix is a piece of cake (even if you were to solve the characteristic polynomial), so you should have no problems.

To obtain the largest (first) eigenvalue and the corresponding eigenvector, use the optional second argument as Eigensystem[matrix, 1]. Here's an example (with a smaller matrix to keep the output small):

mat = RandomInteger[{0, 10}, {3, 3}];
{eigVals, eigVecs} = Eigensystem[mat] // N

(* Out[1]= {{21.4725, 6.39644, 0.131054}, {{1.3448, 0.904702, 1.}, 
            {0.547971, -1.99577, 1.}, {-0.935874, -0.127319, 1.}}} *)

{eigVal1, eigVec1} = Eigensystem[mat, 1] // N
(* Out[2]= {{21.4725}, {{1.3448, 0.904702, 1.}}} *)

Answer 2

If the matrix is completely numerical (not symbolic), then Eigenvalues will return eigenvalues by descending magnitude. Therefore Eigenvalues[matrix, 1] will always give the largest eigenvalue and Eigenvector[matrix, 1] will give the corresponding eigenvector. As R.M. said, both can be obtained at the same time using Eigensystem.

There are different numerical methods for obtaining the eigenvector that corresponds to the largest eigenvalue (by magnitude), the most common being power iteration.

If the matrix is symbolic, the eigenvalues/vectors are not ordered (as far as I know).

Answer 3

Re the question as to how Mathematica finds eigenvalues: in the Documentation Center look up "implementation"; you'll find a link to the page tutorial/SomeNotesOnInternalImplementation. And there, in the section Exact Numerical Linear Algebra you'll find the explanation, "Eigenvalues works by interpolating the characteristic polynomial."

Answer 4

Eigensystem will give you the eigenvalues and eigenvectors, the rest is a matter of sorting and extracting that data.

(* Calculates the eigenvector with the largest eigenvalue *)
largestEV[m_] := Sort[
    Transpose@Eigensystem[m],
    #1[[1]] < #2[[1]] &
][[1, 2]]

m = {{1, 1, 0}, {0, 2, 0}, {0, 2, 3}}
largestEV[m]

 {1, 0, 0}