The following are essentially all 3 × 3 invertible defective matrices that have both integer eigenvalues and 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".

• 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 defective matrices with no entries greater than one in absolute value

There are no invertible defective matrices that have these characteristics.

2 × 2 defective matrices with 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, or integer multiples of previous matrices.

Eigenvalues	Singular values	Matrix
2, -2, -2	4, 2, 1	[2 -2 -1; 0 -2 -2; 0 0 -2]
2, 2, -2	4, 2, 1	[2 -2 -1; 0 2 2; 0 0 -2]
2, 2, 2	4, 2, 1	[2 -2 -1; 0 2 2; 0 0 2]
2, 2, -2	4, 2, 1	[2 -2 0; 0 -2 0; -2 1 2]
2, 2, -2	4, 2, 1	[2 -2 0; 0 -2 0; 2 -1 2]
-2, 2, 2	4, 2, 1	[2 -2 0; 0 2 0; -2 1 -2]
2, 2, 2	4, 2, 1	[2 -2 0; 0 2 0; -2 1 2]
-2, 2, 2	4, 2, 1	[2 -2 0; 0 2 0; 2 -1 -2]
2, 2, 2	4, 2, 1	[2 -2 0; 0 2 0; 2 -1 2]
2, -2, -2	4, 2, 1	[2 -2 1; 0 -2 2; 0 0 -2]
2, 2, -2	4, 2, 1	[2 -2 1; 0 2 -2; 0 0 -2]
2, 2, 2	4, 2, 1	[2 -2 1; 0 2 -2; 0 0 2]
2, -2, -2	4, 2, 1	[2 -1 -2; 0 -2 0; 0 -2 -2]
2, 2, -2	4, 2, 1	[2 -1 -2; 0 -2 0; 0 2 2]
2, 2, 2	4, 2, 1	[2 -1 -2; 0 2 0; 0 2 2]
2, 2, -2	4, 2, 1	[2 -1 2; 0 -2 0; 0 -2 2]
2, -2, -2	4, 2, 1	[2 -1 2; 0 -2 0; 0 2 -2]
2, 2, 2	4, 2, 1	[2 -1 2; 0 2 0; 0 -2 2]
-2, 2, 2	4, 2, 1	[2 0 -2; -2 -2 1; 0 0 2]
2, 2, -2	4, 2, 1	[2 0 -2; -2 2 1; 0 0 -2]
2, 2, 2	4, 2, 1	[2 0 -2; -2 2 1; 0 0 2]
-2, 2, 2	4, 2, 1	[2 0 -2; 2 -2 -1; 0 0 2]
2, 2, -2	4, 2, 1	[2 0 -2; 2 2 -1; 0 0 -2]
2, 2, 2	4, 2, 1	[2 0 -2; 2 2 -1; 0 0 2]
-2, -2, 2	4, 2, 1	[2 0 0; -2 -2 0; -1 -2 -2]
-2, -2, 2	4, 2, 1	[2 0 0; -2 -2 0; 1 2 -2]
-2, 2, 2	4, 2, 1	[2 0 0; -2 2 0; -1 2 -2]
2, 2, 2	4, 2, 1	[2 0 0; -2 2 0; -1 2 2]
-2, 2, 2	4, 2, 1	[2 0 0; -2 2 0; 1 -2 -2]
2, 2, 2	4, 2, 1	[2 0 0; -2 2 0; 1 -2 2]
-2, -2, 2	4, 2, 1	[2 0 0; -1 -2 -2; -2 0 -2]
-2, 2, 2	4, 2, 1	[2 0 0; -1 -2 -2; 2 0 2]
-2, 2, 2	4, 2, 1	[2 0 0; -1 -2 2; -2 0 2]
-2, -2, 2	4, 2, 1	[2 0 0; -1 -2 2; 2 0 -2]
2, 2, 2	4, 2, 1	[2 0 0; -1 2 -2; 2 0 2]
2, 2, 2	4, 2, 1	[2 0 0; -1 2 2; -2 0 2]
-2, 2, 2	4, 2, 1	[2 0 0; 1 -2 -2; -2 0 2]
-2, -2, 2	4, 2, 1	[2 0 0; 1 -2 -2; 2 0 -2]
-2, -2, 2	4, 2, 1	[2 0 0; 1 -2 2; -2 0 -2]
-2, 2, 2	4, 2, 1	[2 0 0; 1 -2 2; 2 0 2]
2, 2, 2	4, 2, 1	[2 0 0; 1 2 -2; -2 0 2]
2, 2, 2	4, 2, 1	[2 0 0; 1 2 2; 2 0 2]
-2, -2, 2	4, 2, 1	[2 0 0; 2 -2 0; -1 2 -2]
-2, -2, 2	4, 2, 1	[2 0 0; 2 -2 0; 1 -2 -2]
-2, 2, 2	4, 2, 1	[2 0 0; 2 2 0; -1 -2 -2]
2, 2, 2	4, 2, 1	[2 0 0; 2 2 0; -1 -2 2]
-2, 2, 2	4, 2, 1	[2 0 0; 2 2 0; 1 2 -2]
2, 2, 2	4, 2, 1	[2 0 0; 2 2 0; 1 2 2]
-2, 2, 2	4, 2, 1	[2 0 2; -2 -2 -1; 0 0 2]
2, 2, -2	4, 2, 1	[2 0 2; -2 2 -1; 0 0 -2]
2, 2, 2	4, 2, 1	[2 0 2; -2 2 -1; 0 0 2]
-2, 2, 2	4, 2, 1	[2 0 2; 2 -2 1; 0 0 2]
2, 2, -2	4, 2, 1	[2 0 2; 2 2 1; 0 0 -2]
2, 2, 2	4, 2, 1	[2 0 2; 2 2 1; 0 0 2]
2, 2, -2	4, 2, 1	[2 1 -2; 0 -2 0; 0 -2 2]
2, -2, -2	4, 2, 1	[2 1 -2; 0 -2 0; 0 2 -2]
2, 2, 2	4, 2, 1	[2 1 -2; 0 2 0; 0 -2 2]
2, -2, -2	4, 2, 1	[2 1 2; 0 -2 0; 0 -2 -2]
2, 2, -2	4, 2, 1	[2 1 2; 0 -2 0; 0 2 2]
2, 2, 2	4, 2, 1	[2 1 2; 0 2 0; 0 2 2]
2, -2, -2	4, 2, 1	[2 2 -1; 0 -2 2; 0 0 -2]
2, 2, -2	4, 2, 1	[2 2 -1; 0 2 -2; 0 0 -2]
2, 2, 2	4, 2, 1	[2 2 -1; 0 2 -2; 0 0 2]
2, 2, -2	4, 2, 1	[2 2 0; 0 -2 0; -2 -1 2]
2, 2, -2	4, 2, 1	[2 2 0; 0 -2 0; 2 1 2]
-2, 2, 2	4, 2, 1	[2 2 0; 0 2 0; -2 -1 -2]
2, 2, 2	4, 2, 1	[2 2 0; 0 2 0; -2 -1 2]
-2, 2, 2	4, 2, 1	[2 2 0; 0 2 0; 2 1 -2]
2, 2, 2	4, 2, 1	[2 2 0; 0 2 0; 2 1 2]
2, -2, -2	4, 2, 1	[2 2 1; 0 -2 -2; 0 0 -2]
2, 2, -2	4, 2, 1	[2 2 1; 0 2 2; 0 0 -2]
2, 2, 2	4, 2, 1	[2 2 1; 0 2 2; 0 0 2]

2 × 2 defective matrices with no entries greater than three in absolute value

There are no invertible defective matrices that have these characteristics.

2 × 2 defective matrices with no entries greater than four in absolute value

There are no invertible defective matrices that have these characteristics other than those that are scalar multiples of previously listed matrices.

2 × 2 defective matrices with no entries greater than five in absolute value

There are no invertible defective matrices that have these characteristics.