# [Maxima] eigenvectors in maxima

jason-sage at creativetrax.com jason-sage at creativetrax.com
Sun May 31 00:07:36 CDT 2009

```Robert Dodier wrote:
> Jason, I looked at the eigenvector code and it appears Maxima
> is just smashing all the vectors together, but it is easy to
> distinguish them. I've attached a patch so that eigenvectors
> returns [vals, vecs] where vals is a list of values and their
> multiplicities (just as it was before) and vecs is a list of lists
> of vectors (before it was just a list of vectors).
>
> Here is a sample session with the examples you gave.
> Sorry, it's probably a little messy. Comments, anyone?
>

No one has commented yet.  Can this patch be put into Maxima?  If this
patch is put in maxima eventually, I can add it to the Sage maxima
package now and fix things.  If this patch is not put into maxima
eventually, then I probably will have resistance to adding it to the
sage package (thus creating a sage-specific fork of maxima, essentially).

Thanks,

Jason

> (%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]);
>                          [ 0  1  0  0 ]
>                          [            ]
>                          [ 0  0  0  0 ]
> (%o1)                    [            ]
>                          [ 0  0  2  0 ]
>                          [            ]
>                          [ 0  0  0  2 ]
> (%i2) [vals, vecs] : eigenvectors (M1);
> (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
>                                    [[0, 0, 1, 0], [0, 0, 0, 1]]]]
> (%i3) for i thru length (vals) do disp (val[i] = vals[i][1], mult[i] =
> vals[i][2], vec[i] = vecs[i]);
>                             val  = 0
>                                1
>
>                             mult  = 2
>                                 1
>
>                       vec  = [[1, 0, 0, 0]]
>                          1
>
>                             val  = 2
>                                2
>
>                             mult  = 2
>                                 2
>
>                vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
>                   2
>
> (%o3)                         done
> (%i4) M2 : matrix ([0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 2, 1], [0, 0, 0, 2]);
>                          [ 0  0  0  0 ]
>                          [            ]
>                          [ 0  0  0  0 ]
> (%o4)                    [            ]
>                          [ 0  0  2  1 ]
>                          [            ]
>                          [ 0  0  0  2 ]
> (%i5) [vals, vecs] : eigenvectors (M2);
> (%o5) [[[0, 2], [2, 2]], [[[1, 0, 0, 0], [0, 1, 0, 0]],
>                                                  [[0, 0, 1, 0]]]]
> (%i6) for i thru length (vals) do disp (val[i] = vals[i][1], mult[i] =
> vals[i][2], vec[i] = vecs[i]);
>                             val  = 0
>                                1
>
>                             mult  = 2
>                                 1
>
>                vec  = [[1, 0, 0, 0], [0, 1, 0, 0]]
>                   1
>
>                             val  = 2
>                                2
>
>                             mult  = 2
>                                 2
> (%i1) M1 : matrix ([0, 1, 0, 0], [0, 0, 0, 0], [0, 0, 2, 0], [0, 0, 0, 2]);
>                          [ 0  1  0  0 ]
>                          [            ]
>                          [ 0  0  0  0 ]
> (%o1)                    [            ]
>                          [ 0  0  2  0 ]
>                          [            ]
>                          [ 0  0  0  2 ]
> (%i2) [vals, vecs] : eigenvectors (M1);
> (%o2) [[[0, 2], [2, 2]], [[[1, 0, 0, 0]],
>                                    [[0, 0, 1, 0], [0, 0, 0, 1]]]]
> (%i3) for i thru length (vals) do disp (val[i] = vals[i][1], mult[i] =
> vals[i][2], vec[i] = vecs[i]);
>                             val  = 0
>                                1
>
>                             mult  = 2
>                                 1
>
>                       vec  = [[1, 0, 0, 0]]
>                          1
>
>                             val  = 2
>                                2
>
>                             mult  = 2
>                                 2
>
>                vec  = [[0, 0, 1, 0], [0, 0, 0, 1]]
>                   2
>
> (%o3)                         done
> (%i4) M2 : matrix ([0, 0, 0, 0], [0, 0, 0, 0], [0, 0, 2, 1], [0, 0, 0, 2]);
>                          [ 0  0  0  0 ]
>                          [            ]
>                          [ 0  0  0  0 ]
> (%o4)                    [            ]
>                          [ 0  0  2  1 ]
>                          [            ]
>                          [ 0  0  0  2 ]
> (%i5) [vals, vecs] : eigenvectors (M2);
> (%o5) [[[0, 2], [2, 2]], [[[1, 0, 0, 0], [0, 1, 0, 0]],
>                                                  [[0, 0, 1, 0]]]]
> (%i6) for i thru length (vals) do disp (val[i] = vals[i][1], mult[i] =
> vals[i][2], vec[i] = vecs[i]);
>                             val  = 0
>                                1
>
>                             mult  = 2
>                                 1
>
>                vec  = [[1, 0, 0, 0], [0, 1, 0, 0]]
>                   1
>
>                             val  = 2
>                                2
>
>                             mult  = 2
>                                 2
>
>                       vec  = [[0, 0, 1, 0]]
>                          2
>
> (%o6)                         done
>
>                       vec  = [[0, 0, 1, 0]]
>                          2
>
> (%o6)                         done
>
>
> FWIW
>
> Robert Dodier
>
> PS.
> --- share/matrix/eigen.mac      22 Nov 2007 17:11:14 -0000      1.4
> +++ share/matrix/eigen.mac      28 May 2009 04:09:26 -0000
> @@ -114,7 +114,7 @@
>                 unknowns:endcons(concat(notknwn,index1),unknowns),
>                 vectr:columnvector(unknowns),matrx:mat.vectr,
>                 nondiagonalizable:false,
> -               listall:[eigvals],realonly:false,algebraic:true,
> +               eigvecs:[],realonly:false,algebraic:true,
>                 for index1 thru count do
>                 (count:mi(count-part(multiplicities,index1)+1),
>                 mmatrx:matrx-part(eigvals,1,index1)*vectr,
> @@ -132,8 +132,8 @@
>                 (print(" "),print("algsys failure: the eigenvector(s) for the",
>                 index1,"th eigenvalue will be missing.")),
>                 if hermitianmatrix and %rnum>1 then unit:gramschmidt(unit),
> -               listall:append(listall,unit)),
> -               return(listall)))\$
> +               eigvecs:append(eigvecs,[unit])),
> +               return([eigvals, eigvecs])))\$
>
>
>  /* the first arg is of the form [r1,r2,r3].
>
>

```