The following are essentially all 2 × 2 defective matrices that have 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 matrix that has two repeated singular values, you can search for, for example, "1, 1".
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.
These are all such matrices up to multiplication by -1, in which case, the singular values are unchanged.
Singular values | Matrix |
---|---|
2, 0 | [ 1 -1; 1 -1] |
2, 0 | [ 1 1; -1 -1] |
These are no such matrices that are not previous matrices multiplied by two, in which case the singular values are also multiplied by two.
These are no such matrices that are not previous matrices multiplied by three, in which case the singular values are also multiplied by three.
These are no such matrices that are not previous matrices multiplied by four, in which case the singular values are also multiplied by four.
These are no such matrices that are not previous matrices multiplied by five, in which case the singular values are also multiplied by five.