Integer matrices with integer eigenvalues
Over 23000 matrices with integer entries and eigenvalues.
When looking for examples to give to students with respect to eigenvalues and eigenvectors,
it is nice to have matrices that have integer eigenvalues, as then both the eigenvalues and the
corresponding eigenvectors can be easily found by students.
There are two approaches taken:
- an exhaustive list of all matrices that satisfy a given set of properties
(for example, symmetric 3 × 3 and invertible with no entries greater
than one in absolute value), and
- randomly generated lists of matrices that satisfy a given set of properties
(for example, non-symmetric 5 × 5 and singular with no entries
greater than five in absolute value).
In the first case, all matrices are listed up to multiplication by -1, so as to reduce
the list by half. Any matrix in the first category can be multiplied by -1 and the corresponding
eigenvalues would, too, be negated. In the second case, only one example is given for each
set of eigenvalues. The reason that random matrices are used to find matrices that have
a unique set of eigenvalues is so that there is no systematic
approach that leads to similar examples with too much repetition in the patterns.
2 × 2 real matrices
3 × 3 real matrices
4 × 4 real matrices
5 × 5 real matrices
2 × 2 complex matrices
3 × 3 complex matrices
4 × 4 complex matrices
5 × 5 complex matrices
See also
nice eigenvalue decompositions (eigendecompositions)
nice singular-value decompositions (SVDs)
integer matrices with integer eigenvalues
integer matrices with integer singular values
integer-normed vectors
integer-normed complex vectors
geometric sequences with nice norms