# Extracting Matrices from the Output of eigenvectors()

In class I sometimes need to use matrices of eigenvalues and eigenvectors, but the output of eigenvectors() isn’t particularly helpful for that out of the box.

Here are two one-liners that work in the case of simple eigenvalues.  I’ll post updates as needed:

First eigU(), takes the output of eigenvectors() and returns matrix of eigenvectors:

`eigU(v):=transpose(apply('matrix,makelist(part(v,2,i,1),i,1,length(part(v,2)),1)));`

And eigdiag(), which takes the output of eigenvectors() and returns diagonal matrix of eigenvalues:

`eigdiag(v):=apply('diag_matrix,part(v,1,1));`

For a matrix with a full set of eigenvectors but eigenvalues of multiplicity greater than one, the lines above fail.  A version of the above that works correctly in that case could look like:

```eigdiag(v):=apply('diag_matrix,flatten(makelist(makelist(part(v,1,1,j),i,1,part(v,1,2,j)),j,1,length(part(v,1,1)))));

eigU(v):=transpose(apply('matrix,makelist(makelist(flatten(part(v,2))[i],i,lsum(i,i,part(v,1,2))*j+1,lsum(i,i,part(v,1,2))*j+lsum(i,i,part(v,1,2))),j,0,lsum(i,i,part(v,1,2))-1)));```

## 5 thoughts on “Extracting Matrices from the Output of eigenvectors()”

1. Larry Lee says:

Hi,
Thanks for the post describing how to use Maxima’s eigenvector and value functions. Unfortunately, there’s still a subtlety in Maxima’s output format that causes the macros you gave above to fail – that is, when Maxima returns vectors that have multiplicities greater than 1.

For example, consider the matrix:
M : matrix (
[1, kb, 0, 0],
[ka, 1, 0, 0],
[0, kb * (1 – kb), 1, 0],
[ka * (1 – ka), 0, 0, 1]
);

Maxima returns three eigenvalues for this matrix and lists the last as having two multiplicities. This breaks your macro.

Do you have any alternate versions that support multiplicities?

Thanks,
Larry

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1. Right!

For a matrix like you describe, I think you could use the lines below
I know they look awful and don’t really fit the description of “quick one-liner”

eigdiag(v):=apply(‘diag_matrix,flatten(makelist(makelist(part(v,1,1,j),i,1,part(v,1,2,j)),j,1,length(part(v,1,1)))));

and

eigU(v):=transpose(apply(‘matrix,makelist(makelist(flatten(part(v,2))[i],i,lsum(i,i,part(v,1,2))*j+1,lsum(i,i,part(v,1,2))*j+lsum(i,i,part(v,1,2))),j,0,lsum(i,i,part(v,1,2))-1)));

These aren’t entirely general either—
This fails in the event that maxima can’t find the eigenvectors, as with
A:matrix([3,6,9],
[2,4,5],
[3,2,1]);

and also fails in the case of a jordan block where there isn’t a full set of eigenvectors
A:matrix([1,1],[0,1])

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2. Markus says:

Found this while working with STACK for Moodle, can confirm that your code works there as well. Thank you very much, I was looking for something like this to create online excercises on diagonalization of (symmetric) matrices.

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