The alt_eigen package

Introduction

The Maxima language alt_eigen package expresses eigenvectors in terms of eigenvalues.

To use this package, you will need to load it manually. Assuming the file alt_eigen.mac is in a path that Maxima can find, load the package using the command

 load("alt_eigen.mac")$

Examples

For our first example, the characteristic polynomial of matrix([1 , 2],[3 ,-4]) factors over the gaussian integers into a product of first degree terms. So for this matrix, the command alt_eigen explicitly finds each eigenvalue and eigenspace:

(%i3) M : matrix([1,2],[3,-4]);
(%o3) matrix([1,2],[3,-4])

(%i4) alt_eigen(M,'var = z, 'maxdegree = 1);
(%o4) [[z=-5,sspan(matrix([-1],[3]))], [z=2,sspan(matrix([-2],[-1]))]]

The last two arguments to alt_eigen are optional and can be in any order. The argument 'var = z tells Maxima to use the name z for the eigenvalue, and the argument maxdegree = 1 instructs the code to solve every factor of the characteristic polynomial that has degree at most one, where the characteristic polynomial is factored over the gaussian integers.

The output is a list of lists, with each list member of the form [eigenvalue, eigenspace], where eigenvalue is a polynomial in the variable var that has been solved for its highest power.

Each eigenspace is represented as the span of a list of column vectors. For the span, the alt_eigen package defines a simplifying object sspan. For the same purpose, the linear algebra package represents a span using a nonsimplifying span object.

For this $3 \times 3$ matrix, one factor of the characteristic polynomial has degree two and the other degree one. Thus, when maxdegree = 1, one eigenspace is expressed in terms of the eigenvalue and the other eigenspace is given explicitly. Changing to maxdegree = 2, all three eigenspaces are given explicitly:

(%i6) M : matrix([1,2,3],[4,5,6],[7,8,9])$

(%i7) alt_eigen(M,'var = z, 'maxdegree = 1);
(%o7) [[z^2=15*z+18,sspan(matrix([26-2*z],[2-z],[-22]))], 
         [z=0,sspan(matrix([-1],[2],[-1]))]]

(%i8) alt_eigen(M,'var = z, 'maxdegree = 2);
(%o8) [[z=(3*sqrt(33)+15)/2,sspan(matrix([22-6*sqrt(33)],[-(3*sqrt(33))-11],[-22]))],  
        [z=-((3*sqrt(33)-15)/2),sspan(matrix([6*sqrt(33)+22],[3*sqrt(33)-11],[-22]))],
  [z=0,sspan(matrix([-1],[2],[-1]))]]

When required, the output is enclosed an assuming object that includes assumptions about parameters that are necessary for the validity of the answer. Especially when used non interactively, a user will need to check if the output is an assuming object.

The matrix entries can be symbolic; here is an example

 (%i1) M : matrix([1,2],[-1/8,q])$
 (%i2) xxx : alt_eigen(M,'var = z,'maxdegree = 1);
 (%o2) assuming(notequal(-(64*(q-2)*q),0),[[z^2=-((-(4*q*z)-4*z+4*q+1)/4),
    sspan(matrix([2],[z-1]))]])

Maxima determined that $-64(q-2) q = 0$ is a special case. Substituting $q=2$ into the above yields unknown:

(%i3) subst(q = 2,xxx);
(%o3) unknown

To find the eigenspace when $q = 2$, you must substitute $q = 2$ before calling alt_eigen. But if $q \neq 2$, say $q = 5$, we can make that substitution in the previous result and find that

(%i5) subst(q=5,xxx);
(%o5) [[z^2=-((21-24*z)/4),sspan(matrix([2],[z-1]))]]

For an eigenspace with dimension two or greater, we can optionally apply the Gram-Schmidt process to find an orthogonal basis for the eigenspace. Here is an example

(%i2) M : matrix([15714, 24872, 12436], [-1450, -2151, -1160],[-7025, -11240, -5451])$

(%i3) alt_eigen(M,'var = z);
(%o3) [z=169,[matrix([-8],[5],[0]),matrix([0],[1],[-2])],
         z=7774,[matrix([3109],[-290],[-1405])]]

To apply Gram-Schmidt to the eigenspace, use the optional argument 'orthogonal = true

(%i1) alt_eigen(M,'var = z,'orthogonal = true);
(%o1) [[z=169,sspan(matrix([-8],[5],[0]),
             matrix([20],[32],[-89]))],
   [z=7774,sspan(matrix([3109],[-290],[-1405]))]]

We end with a $5\times 5$ example. For this example, we anticipate that $q = 0$ will be a special case, we start with issuing the Maxima command assume(notequal(q,0)):

(%i4) M : genmatrix(lambda([a,b], if a=b then 1 elseif abs(a-b)=1 then -q else 0),5,5);
(%o4) matrix([1,-q,0, 0,0],
   [-q,1,-q,0,0],
   [0,-q,1,-q,0],
   [0,0,-q,1,-q],
   [0,0,0,-q,1])
(%i5) assume(notequal(q,0))$

(%i6) alt_eigen(M, 'var = z,maxdegree = 1);
(%o6) [[z^2=2*z+3*q^2-1,sspan(matrix([-q],[z-1],[-2*q],[z-1],[-q]))],
         [z=1-q,sspan(matrix([1],[1],[0],[-1],[-1]))],
   [z=q+1,sspan(matrix([1],[-1],[0],[1],[-1]))],
   [z=1,sspan(matrix([-1],[0],[1],[0],[-1]))]]

Testing

Assuming the file alt_eigen.mac is in a path that Maxima can find, to run the tests for this package, issue the command

 batch(rtest_alt_eigen,'test);

Maxima version 5.47.0 should complete all the tests with no failures. If you find that some tests fail, please report them to the alt_eigen discussion list.

Name of this package

The package name alt_eigen isn't particularly descriptive. As you likely suspect, the alt in the package name is a shortening of the word alternative, meaning an alternative to the standard Maxima package for eigenvalues and eigenvectors.