3 × 3 invertible matrices with non-real eigenvalues
The following are essentially all 3 × 3 invertible matrices that have non-real eigenvalues but integer singular values. They are grouped based on the maximum integer in absolute value in the matrix. The singular values are sorted, so if you want an invertible matrix that has two repeated singular values, you can search for, for example, "1, 1".
- No entries greater than one in absolute value
- No entries greater than two in absolute value
- No entries greater than three in absolute value
- No entries greater than four in absolute value
- No entries greater than five in absolute value
While the matrices are in the Matlab format, some of these have been tested in Maple to ensure that they are not the result of numeric error.
3 × 3 invertible matrices with non-real eigenvalues and no entries greater than one in absolute value
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged.
| Eigenvalues | Singular values | Matrix |
|---|---|---|
| -j, j, -1 | 1, 1, 1 | [0 0 1; 0 -1 0; -1 0 0] |
| -j, j, 1 | 1, 1, 1 | [0 0 1; 0 1 0; -1 0 0] |
| -j, j, -1 | 1, 1, 1 | [0 1 0; -1 0 0; 0 0 -1] |
| -j, j, 1 | 1, 1, 1 | [0 1 0; -1 0 0; 0 0 1] |
| -j, j, 1 | 1, 1, 1 | [1 0 0; 0 0 -1; 0 1 0] |
| -j, j, 1 | 1, 1, 1 | [1 0 0; 0 0 1; 0 -1 0] |
3 × 3 invertible matrices with non-real eigenvalues and no entries greater than two in absolute value
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged.
| Eigenvalues | Singular values | Matrix |
|---|---|---|
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 -2 -2; -2 1 2; 2 -2 -1] |
| -1 - 2j, -1 + 2j, -1 | 5, 1, 1 | [1 -2 -2; 2 -2 -1; 2 -1 -2] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 -2 -2; 2 -1 -2; -2 2 1] |
| -1 - 2j, -1 + 2j, 1 | 5, 1, 1 | [1 -2 -2; 2 -1 -2; 2 -2 -1] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 -2 -1; -2 -2 -2; 1 2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 -2 -1; 2 2 2; 1 2 -1] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 -2 1; -2 -2 2; -1 -2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 -2 1; 2 2 -2; -1 -2 -1] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 -2 2; -2 1 -2; -2 2 -1] |
| -1 - 2j, -1 + 2j, -1 | 5, 1, 1 | [1 -2 2; 2 -2 1; -2 1 -2] |
| -1 - 2j, -1 + 2j, 1 | 5, 1, 1 | [1 -2 2; 2 -1 2; -2 2 -1] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 -2 2; 2 -1 2; 2 -2 1] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 -1 -2; 1 -1 2; -2 -2 -2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 -1 -2; 1 -1 2; 2 2 2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 -1 2; 1 -1 -2; -2 -2 2] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 -1 2; 1 -1 -2; 2 2 -2] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 1 -2; -1 -1 -2; -2 2 -2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 1 -2; -1 -1 -2; 2 -2 2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 1 2; -1 -1 2; -2 2 2] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 1 2; -1 -1 2; 2 -2 -2] |
| -1 - 2j, -1 + 2j, -1 | 5, 1, 1 | [1 2 -2; -2 -2 1; 2 1 -2] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 2 -2; -2 -1 2; -2 -2 1] |
| -1 - 2j, -1 + 2j, 1 | 5, 1, 1 | [1 2 -2; -2 -1 2; 2 2 -1] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 2 -2; 2 1 -2; 2 2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 2 -1; -2 2 -2; 1 -2 -1] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 2 -1; 2 -2 2; 1 -2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [1 2 1; -2 2 2; -1 2 -1] |
| 2 -2 - 2j, -2 + 2j, | 4, 2, 2 | [1 2 1; 2 -2 -2; -1 2 -1] |
| -1 - 2j, -1 + 2j, -1 | 5, 1, 1 | [1 2 2; -2 -2 -1; -2 -1 -2] |
| -1 - 2j, -1 + 2j, 1 | 5, 1, 1 | [1 2 2; -2 -1 -2; -2 -2 -1] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 2 2; -2 -1 -2; 2 2 1] |
| -1, 1 - 2j, 1 + 2j | 5, 1, 1 | [1 2 2; 2 1 2; -2 -2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 -2 -2; -2 -1 1; 2 -1 1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 -2 -2; 2 1 -1; -2 1 -1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 -2 -1; 2 -1 -2; -1 2 2] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 -2 1; 2 -1 2; 1 -2 2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 -2 2; -2 -1 -1; -2 1 1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 -2 2; 2 1 1; 2 -1 -1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 -1 -2; -1 2 2; 2 -2 -1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 -1 2; -1 2 -2; -2 2 -1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 1 -2; 1 2 -2; 2 2 -1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 1 2; 1 2 2; -2 -2 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 2 -2; -2 1 1; -2 -1 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 2 -2; 2 -1 -1; 2 1 1] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 2 -1; -2 -1 2; -1 -2 2] |
| 1 - 2j, 1 + 2j, 1 | 5, 1, 1 | [2 2 1; -2 -1 -2; 1 2 2] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 2 2; -2 1 -1; 2 1 -1] |
| 2 - 2j, 2 + 2j, -2 | 4, 2, 2 | [2 2 2; 2 -1 1; -2 -1 1] |
3 × 3 invertible matrices with non-real eigenvalues and no entries greater than three in absolute value
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged. This does not include integer multiples of matrices listed above.
| Eigenvalues | Singular values | Matrix |
|---|---|---|
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 -3 -2; -3 2 -2; 2 2 3] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 -3 2; -3 2 2; -2 -2 3] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 -2 -3; 2 3 2; -3 -2 2] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 -2 3; 2 3 -2; 3 2 2] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 2 -3; -2 3 -2; -3 2 2] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 2 3; -2 3 2; 3 -2 2] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 3 -2; 3 2 2; 2 -2 3] |
| 5, 1 - 2j, 1 + 2j | 5, 5, 1 | [2 3 2; 3 2 -2; -2 2 3] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 -2 -2; -2 3 -2; 2 2 -3] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 -2 -2; 2 -3 2; -2 -2 3] |
| 1 - 2j, 1 + 2j, -5 | 5, 5, 1 | [3 -2 -2; 2 -3 2; 2 2 -3] |
| 1 - 2j, 1 + 2j, 5 | 5, 5, 1 | [3 -2 -2; 2 2 -3; 2 -3 2] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 -2 2; -2 3 2; -2 -2 -3] |
| 1 - 2j, 1 + 2j, -5 | 5, 5, 1 | [3 -2 2; 2 -3 -2; -2 -2 -3] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 -2 2; 2 -3 -2; 2 2 3] |
| 1 - 2j, 1 + 2j, 5 | 5, 5, 1 | [3 -2 2; 2 2 3; -2 3 2] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 2 -2; -2 -3 -2; -2 2 3] |
| 1 - 2j, 1 + 2j, -5 | 5, 5, 1 | [3 2 -2; -2 -3 -2; 2 -2 -3] |
| 1 - 2j, 1 + 2j, 5 | 5, 5, 1 | [3 2 -2; -2 2 3; 2 3 2] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 2 -2; 2 3 2; 2 -2 -3] |
| 1 - 2j, 1 + 2j, -5 | 5, 5, 1 | [3 2 2; -2 -3 2; -2 2 -3] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 2 2; -2 -3 2; 2 -2 3] |
| 1 - 2j, 1 + 2j, 5 | 5, 5, 1 | [3 2 2; -2 2 -3; -2 -3 2] |
| 5 -1 - 2j, -1 + 2j, | 5, 5, 1 | [3 2 2; 2 3 -2; -2 2 -3] |
3 × 3 invertible matrices with non-real eigenvalues and no entries greater than four in absolute value
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged. This does not include integer multiples of matrices listed above.
| Eigenvalues | Singular values | Matrix |
|---|---|---|
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 -4 -2; -2 -2 4; -4 1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -4 -2; -2 2 -4; 4 1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -4 -2; 2 -2 4; -4 -1 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 -4 2; -2 -2 -4; 4 -1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -4 2; -2 2 4; -4 -1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -4 2; 2 -2 -4; 4 1 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 -2 -4; -4 -2 1; -2 4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -2 -4; -4 2 -1; 2 4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -2 -4; 4 -2 1; -2 -4 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 -2 4; -4 -2 -1; 2 -4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -2 4; -4 2 1; -2 -4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 -2 4; 4 -2 -1; 2 4 2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 2 -4; -4 -2 -1; -2 4 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 2 -4; 4 -2 -1; -2 -4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 2 -4; 4 2 1; 2 -4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 2 4; -4 -2 1; 2 -4 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 2 4; 4 -2 1; 2 4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 2 4; 4 2 -1; -2 4 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 4 -2; -2 -2 -4; -4 1 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 4 -2; 2 -2 -4; -4 -1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 4 -2; 2 2 4; 4 -1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 4 2; -2 -2 4; 4 -1 2] |
| 5 -4 - 2j, -4 + 2j, | 5, 5, 4 | [1 4 2; 2 -2 4; 4 1 -2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [1 4 2; 2 2 -4; -4 1 -2] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 -4 -4; -4 -4 2; 2 -1 2] |
| 6 -2 - 2j, -2 + 2j, | 8, 6, 1 | [2 -4 -4; -4 2 -4; -3 4 -2] |
| 6 -2 - 2j, -2 + 2j, | 8, 6, 1 | [2 -4 -4; -4 2 -4; 4 -3 -2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 -4 -4; -4 2 3; 4 4 -2] |
| 6 -2 - 2j, -2 + 2j, | 8, 6, 1 | [2 -4 -4; -3 -2 4; -4 -4 2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 -4; -3 -2 4; 4 4 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 -4; -3 4 -2; 4 -2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 -4 -4; 2 2 -1; -4 2 -4] |
| 6 -2 - 2j, -2 + 2j, | 8, 6, 1 | [2 -4 -4; 4 -2 -3; -4 -4 2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 -4 -4; 4 -2 4; -4 3 2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 -4; 4 -2 4; -3 4 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 -4; 4 4 -2; -3 -2 4] |
| 6 -2 - 2j, -2 + 2j, | 8, 6, 1 | [2 -4 -3; -4 2 4; -4 -4 -2] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 -4 -3; 4 -2 -4; -4 -4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 -4 -3; 4 4 2; -4 2 4] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -4 -2; -1 2 -4; -4 -2 -1] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [2 -4 -2; 1 -2 4; -4 -2 1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -4 -2; 1 -2 4; 4 2 -1] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -4 -1; -2 -1 -4; -4 -2 2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [2 -4 -1; -2 1 -4; 4 2 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -4 -1; 2 -1 4; 4 2 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -4 1; -2 -1 4; 4 2 2] |
| 5 -2 - 4j, -2 + 4j, | 5, 5, 4 | [2 -4 1; -2 1 4; -4 -2 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -4 1; 2 -1 -4; -4 -2 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -4 2; -1 2 4; 4 2 -1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -4 2; 1 -2 -4; -4 -2 -1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 -4 2; 1 -2 -4; 4 2 1] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -4 3; -4 2 -4; 4 4 -2] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 -4 3; 4 -2 4; 4 4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 -4 3; 4 4 -2; 4 -2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 -4 4; -4 -4 -2; -2 1 2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 -4 4; -4 2 -3; -4 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -4 4; -4 2 4; -4 3 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -4 4; -4 2 4; 3 -4 -2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 4; -3 -2 -4; -4 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -4 4; -3 -2 -4; 4 4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 4; -3 4 2; -4 2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 -4 4; 2 2 1; 4 -2 -4] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 4; 4 -2 -4; 3 -4 -2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 -4 4; 4 -2 -4; 4 -3 2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -4 4; 4 -2 3; 4 4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 -4 4; 4 4 2; 3 2 4] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -3 -4; -4 -2 -4; -4 4 2] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 -3 -4; -4 -2 -4; 4 -4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 -3 -4; -4 4 2; 4 2 4] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 -3 4; -4 -2 4; -4 4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 -3 4; -4 -2 4; 4 -4 2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 -3 4; -4 4 -2; -4 -2 4] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -2 -4; -4 -1 -2; -1 -4 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 -2 -4; -4 1 -2; 1 4 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -2 -4; 4 -1 2; 1 4 -2] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 -2 -1; 4 2 4; 2 4 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 -2 -1; 4 4 0; 2 0 4] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 -2 1; 4 2 -4; -2 -4 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 -2 1; 4 4 0; -2 0 4] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -2 4; -4 -1 2; 1 4 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 -2 4; -4 1 2; -1 -4 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -2 4; 4 -1 -2; -1 -4 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -1 -4; -4 2 -2; -2 -4 -1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 -1 -4; 4 -2 2; -2 -4 1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -1 -4; 4 -2 2; 2 4 -1] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 -1 -2; 2 -4 4; 4 4 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 -1 -2; 2 4 0; 4 0 4] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 -1 2; 2 -4 -4; -4 -4 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 -1 2; 2 4 0; -4 0 4] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 -1 4; -4 2 2; 2 4 -1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 -1 4; 4 -2 -2; -2 -4 -1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 -1 4; 4 -2 -2; 2 4 1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 1 -4; -4 -2 -2; -2 4 1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 1 -4; -4 -2 -2; 2 -4 -1] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 1 -4; 4 2 2; -2 4 -1] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 1 -2; -2 -4 -4; 4 -4 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 1 -2; -2 4 0; 4 0 4] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 1 2; -2 -4 4; -4 4 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 1 2; -2 4 0; -4 0 4] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 1 4; -4 -2 2; -2 4 -1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 1 4; -4 -2 2; 2 -4 1] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 1 4; 4 2 -2; 2 -4 -1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 2 -4; -4 -1 -2; 1 -4 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 2 -4; 4 -1 2; -1 4 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 2 -4; 4 1 2; 1 -4 -2] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 2 -1; -4 2 -4; 2 -4 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 2 -1; -4 4 0; 2 0 4] |
| 3 - 3j, 3 + 3j, -6 | 6, 6, 3 | [2 2 1; -4 2 4; -2 4 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [2 2 1; -4 4 0; -2 0 4] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 2 4; -4 -1 2; -1 4 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 2 4; 4 -1 -2; 1 -4 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 2 4; 4 1 -2; -1 4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 3 -4; 4 -2 4; -4 -4 2] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 3 -4; 4 -2 4; 4 4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 3 -4; 4 4 -2; 4 -2 4] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 3 4; 4 -2 -4; -4 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 3 4; 4 -2 -4; 4 4 2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 3 4; 4 4 2; -4 2 4] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 4 -4; -4 -2 -4; -4 -3 2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 -4; -4 -2 -4; -3 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 -4; -4 -2 3; -4 4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 -4; -4 4 2; -3 2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 4 -4; -2 2 1; -4 -2 -4] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 -4; 3 -2 -4; -4 4 2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 -4; 3 -2 -4; 4 -4 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 -4; 3 4 2; 4 2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 4 -4; 4 -4 -2; 2 1 2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 4 -4; 4 2 -3; 4 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 -4; 4 2 4; -3 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 -4; 4 2 4; 4 3 -2] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 4 -3; -4 -2 4; -4 4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 4 -3; -4 4 -2; -4 -2 4] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 -3; 4 2 -4; -4 4 -2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 4 -2; -1 -2 -4; -4 2 1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 4 -2; -1 -2 -4; 4 -2 -1] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 4 -2; 1 2 4; -4 2 -1] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 4 -1; -2 -1 -4; 4 -2 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 4 -1; 2 -1 4; -4 2 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 4 -1; 2 1 4; 4 -2 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 4 1; -2 -1 4; -4 2 -2] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 4 1; 2 -1 -4; 4 -2 2] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 4 1; 2 1 -4; -4 2 -2] |
| 2 - 4j, 2 + 4j, -5 | 5, 5, 4 | [2 4 2; -1 -2 4; -4 2 -1] |
| 5, -2 - 4j, -2 + 4j | 5, 5, 4 | [2 4 2; -1 -2 4; 4 -2 1] |
| -5, 4 - 2j, 4 + 2j | 5, 5, 4 | [2 4 2; 1 2 -4; 4 -2 -1] |
| 2 - 2j, 2 + 2j, -6 | 8, 6, 1 | [2 4 3; -4 -2 -4; 4 -4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [2 4 3; -4 4 2; 4 2 4] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 3; 4 2 4; 4 -4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 4; -4 -2 -3; 4 -4 2] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 4; -4 -2 4; 3 4 -2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 4 4; -4 -2 4; 4 3 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 4; -4 4 -2; 3 -2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 4 4; -2 2 -1; 4 2 -4] |
| -6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 4; 3 -2 4; -4 4 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 4; 3 -2 4; 4 -4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [2 4 4; 3 4 -2; -4 -2 4] |
| -6, 3 - 3j, 3 + 3j | 6, 6, 3 | [2 4 4; 4 -4 2; -2 -1 2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 4; 4 2 -4; -4 -3 -2] |
| 6, -2 - 2j, -2 + 2j | 8, 6, 1 | [2 4 4; 4 2 -4; 3 4 -2] |
| -2 - 2j, -2 + 2j, 6 | 8, 6, 1 | [2 4 4; 4 2 3; -4 4 -2] |
| 3 - 4j, 3 + 4j, -4 | 5, 5, 4 | [3 -4 0; 4 3 0; 0 0 -4] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [3 -4 0; 4 3 0; 0 0 4] |
| 3 - 4j, 3 + 4j, -4 | 5, 5, 4 | [3 0 -4; 0 -4 0; 4 0 3] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [3 0 -4; 0 4 0; 4 0 3] |
| 3 - 4j, 3 + 4j, -4 | 5, 5, 4 | [3 0 4; 0 -4 0; -4 0 3] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [3 0 4; 0 4 0; -4 0 3] |
| 3 - 4j, 3 + 4j, -4 | 5, 5, 4 | [3 4 0; -4 3 0; 0 0 -4] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [3 4 0; -4 3 0; 0 0 4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 -4 -3; 4 -4 0; 3 0 -4] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 -4 -2; -4 -2 -4; 1 2 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -4 -2; -3 2 4; -2 -4 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -4 -2; 4 2 -3; -2 -4 4] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 -4 -2; 4 2 4; -2 3 4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 -4 0; 2 2 -1; 0 2 4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 -4 0; 2 2 1; 0 -2 4] |
| -3j, 3j, 4 | 9, 4, 1 | [4 -4 0; 4 -4 -3; 0 3 4] |
| -3j, 3j, 4 | 9, 4, 1 | [4 -4 0; 4 -4 3; 0 -3 4] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 -4 2; -4 -2 4; -1 -2 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -4 2; -3 2 -4; 2 4 4] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 -4 2; 4 2 -4; 2 -3 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -4 2; 4 2 3; 2 4 4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 -4 3; 4 -4 0; -3 0 -4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 -3 -4; 3 -4 0; 4 0 -4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -3 -2; -4 2 -4; -2 4 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 -3 0; 3 -4 -4; 0 4 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 -3 0; 3 -4 4; 0 -4 4] |
| 4 - 3j, 4 + 3j, -4 | 5, 5, 4 | [4 -3 0; 3 4 0; 0 0 -4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -3 0; 3 4 0; 0 0 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -3 2; -4 2 4; 2 -4 4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 -3 4; 3 -4 0; -4 0 -4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 -4; -2 4 -4; -3 4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 -4; -2 4 -4; 4 -3 2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 -2 -4; -2 4 3; 4 4 2] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 -2 -4; 1 -2 2; -4 -4 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 -3; -2 4 4; -4 -4 2] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 -2; 2 4 -1; 2 1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 -2; 2 4 1; 2 -1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 -1; 2 4 -2; 1 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 -1; 2 4 2; 1 -2 4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 -2 0; 1 2 -2; 0 4 4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 -2 0; 1 2 2; 0 -4 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 1; 2 4 -2; -1 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 1; 2 4 2; -1 -2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 2; 2 4 -1; -2 1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -2 2; 2 4 1; -2 -1 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 3; -2 4 -4; 4 4 2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 -2 4; -2 4 -3; -4 -4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 4; -2 4 4; -4 3 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 -2 4; -2 4 4; 3 -4 2] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 -2 4; 1 -2 -2; 4 4 -2] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -1 -2; 1 4 -2; 2 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -1 -2; 1 4 2; 2 -2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -1 2; 1 4 -2; -2 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 -1 2; 1 4 2; -2 -2 4] |
| -3j, 3j, 4 | 9, 4, 1 | [4 0 -4; 0 4 -3; 4 3 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 0 -4; 0 4 -2; 2 1 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 0 -4; 0 4 2; 2 -1 2] |
| -3j, 3j, 4 | 9, 4, 1 | [4 0 -4; 0 4 3; 4 -3 -4] |
| 4 - 3j, 4 + 3j, -4 | 5, 5, 4 | [4 0 -3; 0 -4 0; 3 0 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 0 -3; 0 4 -4; 3 4 -4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 0 -3; 0 4 0; 3 0 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 0 -3; 0 4 4; 3 -4 -4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 0 -2; 0 4 -4; 1 2 2] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 0 -2; 0 4 4; 1 -2 2] |
| -4 - 3j, -4 + 3j, 4 | 5, 5, 4 | [4 0 0; 0 -4 -3; 0 3 -4] |
| -4 - 3j, -4 + 3j, 4 | 5, 5, 4 | [4 0 0; 0 -4 3; 0 -3 -4] |
| -3 - 4j, -3 + 4j, 4 | 5, 5, 4 | [4 0 0; 0 -3 -4; 0 4 -3] |
| -3 - 4j, -3 + 4j, 4 | 5, 5, 4 | [4 0 0; 0 -3 4; 0 -4 -3] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [4 0 0; 0 3 -4; 0 4 3] |
| 3 - 4j, 3 + 4j, 4 | 5, 5, 4 | [4 0 0; 0 3 4; 0 -4 3] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 0 0; 0 4 -3; 0 3 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 0 0; 0 4 3; 0 -3 4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 0 2; 0 4 -4; -1 2 2] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 0 2; 0 4 4; -1 -2 2] |
| 4 - 3j, 4 + 3j, -4 | 5, 5, 4 | [4 0 3; 0 -4 0; -3 0 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 0 3; 0 4 -4; -3 4 -4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 0 3; 0 4 0; -3 0 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 0 3; 0 4 4; -3 -4 -4] |
| -3j, 3j, 4 | 9, 4, 1 | [4 0 4; 0 4 -3; -4 3 -4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 0 4; 0 4 -2; -2 1 2] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 0 4; 0 4 2; -2 -1 2] |
| -3j, 3j, 4 | 9, 4, 1 | [4 0 4; 0 4 3; -4 -3 -4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 1 -2; -1 4 -2; 2 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 1 -2; -1 4 2; 2 -2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 1 2; -1 4 -2; -2 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 1 2; -1 4 2; -2 -2 4] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 2 -4; -1 -2 -2; -4 4 -2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 2 -4; 2 4 -3; 4 -4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 -4; 2 4 4; -3 -4 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 -4; 2 4 4; 4 3 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 -3; 2 4 -4; -4 4 2] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 -2; -2 4 -1; 2 1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 -2; -2 4 1; 2 -1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 -1; -2 4 -2; 1 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 -1; -2 4 2; 1 -2 4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 2 0; -1 2 -2; 0 4 4] |
| 4, 3 - 3j, 3 + 3j | 6, 4, 3 | [4 2 0; -1 2 2; 0 -4 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 1; -2 4 -2; -1 2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 1; -2 4 2; -1 -2 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 2; -2 4 -1; -2 1 4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 2 2; -2 4 1; -2 -1 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 3; 2 4 4; 4 -4 2] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 2 4; -1 -2 2; 4 -4 -2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 4; 2 4 -4; -4 -3 2] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 2 4; 2 4 -4; 3 4 2] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 2 4; 2 4 3; -4 4 2] |
| -3j, 3j, -4 | 9, 4, 1 | [4 3 -4; -3 -4 0; 4 0 -4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 3 -2; 4 2 4; -2 -4 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 3 0; -3 -4 -4; 0 4 4] |
| 4, -3j, 3j | 9, 4, 1 | [4 3 0; -3 -4 4; 0 -4 4] |
| 4 - 3j, 4 + 3j, -4 | 5, 5, 4 | [4 3 0; -3 4 0; 0 0 -4] |
| 4 - 3j, 4 + 3j, 4 | 5, 5, 4 | [4 3 0; -3 4 0; 0 0 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 3 2; 4 2 -4; 2 4 4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 3 4; -3 -4 0; -4 0 -4] |
| -3j, 3j, -4 | 9, 4, 1 | [4 4 -3; -4 -4 0; 3 0 -4] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 4 -2; -4 2 -4; -2 -3 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 4 -2; -4 2 3; -2 4 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 4 -2; 3 2 -4; -2 4 4] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 4 -2; 4 -2 4; 1 -2 -2] |
| -3j, 3j, 4 | 9, 4, 1 | [4 4 0; -4 -4 -3; 0 3 4] |
| -3j, 3j, 4 | 9, 4, 1 | [4 4 0; -4 -4 3; 0 -3 4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 4 0; -2 2 -1; 0 2 4] |
| 3 - 3j, 3 + 3j, 4 | 6, 4, 3 | [4 4 0; -2 2 1; 0 -2 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 4 2; -4 2 -3; 2 -4 4] |
| 2 - 2j, 2 + 2j, 6 | 8, 6, 1 | [4 4 2; -4 2 4; 2 3 4] |
| 6, 2 - 2j, 2 + 2j | 8, 6, 1 | [4 4 2; 3 2 4; 2 -4 4] |
| 6, -3 - 3j, -3 + 3j | 6, 6, 3 | [4 4 2; 4 -2 -4; -1 2 -2] |
| -3j, 3j, -4 | 9, 4, 1 | [4 4 3; -4 -4 0; -3 0 -4] |
3 × 3 invertible matrices with non-real eigenvalues and no entries greater than five in absolute value
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged. This does not include integer multiples of matrices listed above.
| Eigenvalues | Singular values | Matrix |
|---|---|---|
| -4j, 4j, -5 | 8, 5, 2 | [3 -5 0; 5 -3 0; 0 0 -5] |
| -4j, 4j, 5 | 8, 5, 2 | [3 -5 0; 5 -3 0; 0 0 5] |
| -4j, 4j, -5 | 8, 5, 2 | [3 0 -5; 0 -5 0; 5 0 -3] |
| -4j, 4j, 5 | 8, 5, 2 | [3 0 -5; 0 5 0; 5 0 -3] |
| -4j, 4j, -5 | 8, 5, 2 | [3 0 5; 0 -5 0; -5 0 -3] |
| -4j, 4j, 5 | 8, 5, 2 | [3 0 5; 0 5 0; -5 0 -3] |
| -4j, 4j, -5 | 8, 5, 2 | [3 5 0; -5 -3 0; 0 0 -5] |
| -4j, 4j, 5 | 8, 5, 2 | [3 5 0; -5 -3 0; 0 0 5] |
| -3j, 3j, -5 | 9, 5, 1 | [4 -5 0; 5 -4 0; 0 0 -5] |
| -3j, 3j, 5 | 9, 5, 1 | [4 -5 0; 5 -4 0; 0 0 5] |
| -3j, 3j, -5 | 9, 5, 1 | [4 0 -5; 0 -5 0; 5 0 -4] |
| -3j, 3j, 5 | 9, 5, 1 | [4 0 -5; 0 5 0; 5 0 -4] |
| -3j, 3j, -5 | 9, 5, 1 | [4 0 5; 0 -5 0; -5 0 -4] |
| -3j, 3j, 5 | 9, 5, 1 | [4 0 5; 0 5 0; -5 0 -4] |
| -3j, 3j, -5 | 9, 5, 1 | [4 5 0; -5 -4 0; 0 0 -5] |
| -3j, 3j, 5 | 9, 5, 1 | [4 5 0; -5 -4 0; 0 0 5] |
| -3j, 3j, 5 | 9, 5, 1 | [5 0 0; 0 -4 -5; 0 5 4] |
| -3j, 3j, 5 | 9, 5, 1 | [5 0 0; 0 -4 5; 0 -5 4] |
| -4j, 4j, 5 | 8, 5, 2 | [5 0 0; 0 -3 -5; 0 5 3] |
| -4j, 4j, 5 | 8, 5, 2 | [5 0 0; 0 -3 5; 0 -5 3] |
| -4j, 4j, 5 | 8, 5, 2 | [5 0 0; 0 3 -5; 0 5 -3] |
| -4j, 4j, 5 | 8, 5, 2 | [5 0 0; 0 3 5; 0 -5 -3] |
| -3j, 3j, 5 | 9, 5, 1 | [5 0 0; 0 4 -5; 0 5 -4] |
| -3j, 3j, 5 | 9, 5, 1 | [5 0 0; 0 4 5; 0 -5 -4] |
Douglas Wilhelm Harder
Department of Electrical and Computer Engineering
University of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1
Phone: +1 519 888 4567 extension 37023
Fax: +1 519 746 3077
Department of Electrical and Computer Engineering
University of Waterloo
200 University Avenue West
Waterloo, Ontario, Canada N2L 3G1
Phone: +1 519 888 4567 extension 37023
Fax: +1 519 746 3077
Website maintained by Douglas Wilhelm Harder