The following are essentially all 2 × 2 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.

2 × 2 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.

Singular values	Matrix
1, 1	[ 0  1; -1  0]

2 × 2 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.

Singular values	Matrix
1, 3	[ 1 -2;  2 -1]
1, 3	[ 1  2; -2 -1]
1, 3	[ 2 -1;  1 -2]
1, 3	[ 2  1; -1 -2]
2, 3	[ 1 -2;  2  2]
2, 3	[ 1  2; -2  2]
2, 3	[ 2 -2; -1 -2]
2, 3	[ 2 -2;  1  2]
2, 3	[ 2 -2;  2  1]
2, 3	[ 2 -1; -2 -2]
2, 3	[ 2 -1;  2  2]
2, 3	[ 2  1; -2  2]
2, 3	[ 2  1;  2 -2]
2, 3	[ 2  2; -2  1]
2, 3	[ 2  2; -1  2]
2, 3	[ 2  2;  1 -2]

2 × 2 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.

Singular values	Matrix
1, 5	[ 2 -3;  3 -2]
1, 5	[ 2  3; -3 -2]
1, 5	[ 3 -2;  2 -3]
1, 5	[ 3  2; -2 -3]
2, 4	[ 1 -3;  3 -1]
2, 4	[ 1  3; -3 -1]
2, 4	[ 3 -1;  1 -3]
2, 4	[ 3  1; -1 -3]

2 × 2 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.

Singular values	Matrix
1, 7	[ 3 -4;  4 -3]
1, 7	[ 3  4; -4 -3]
1, 7	[ 4 -3;  3 -4]
1, 7	[ 4  3; -3 -4]
3, 5	[ 1 -4;  4 -1]
3, 5	[ 1  4; -4 -1]
3, 5	[ 4 -1;  1 -4]
3, 5	[ 4  1; -1 -4]
5, 5	[ 3 -4;  4  3]
5, 5	[ 3  4; -4  3]
5, 5	[ 4 -3;  3  4]
5, 5	[ 4  3; -3  4]

2 × 2 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.

Singular values	Matrix
1, 9	[ 4 -5;  5 -4]
1, 9	[ 4  5; -5 -4]
2, 8	[ 3 -5;  5 -3]
2, 8	[ 3  5; -5 -3]
3, 7	[ 2 -5;  5 -2]
3, 7	[ 2  5; -5 -2]
3, 7	[ 5 -2;  2 -5]
3, 7	[ 5  2; -2 -5]
4, 6	[ 1 -5;  5 -1]
4, 6	[ 1  5; -5 -1]
4, 6	[ 5 -1;  1 -5]
4, 6	[ 5  1; -1 -5]