Eigenvalues and Eigenvectors with Maple

> with(linalg):

This example has complex eingenvalues.

> A1:=matrix([[1,-2],[1,-1]]);

> eigenvalues(A1);

> eigenvectors(A1);

The output format is as follows: [eigenvalue, multiplicity, { basis for eigenspace } ]. Note: the output is in the form of row vectors to save space; the eigenvectors are really column vectors.

This example has three real distinct eigenvalues. This means A can be diagonalized.

> A2:=matrix([[1,2,1],[2,1,1],[1,1,0]]);

> eigenvalues(A2);

> eigenvectors(A2);

Write this out by hand as. It is a bit hard on the eyes.

This next example has two real distinct eigenvaules. But the eigenspace of 2 is only one dimensional. Hence this matrix cannot be diagonalized.

> A3:=matrix([[2,1,-4],[0,1,3],[0,0,2]]);

> eigenvalues(A3);

> eigenvectors(A3);

This matrix, below, has only two distinct real eigenvaules, but the eigenspaces have full dimension.

> A4:=matrix([[1,2,5],[0,2,0],[0,0,2]]);

> eigenvalues(A4);

> eigenvectors(A4);

Maple does not have a diagonalize command. Instead it has a command called Jordan. Study it and see what it does to the matrices above.