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)));

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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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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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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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