Skip to the content of the web site.

3 × 3 invertible complex matrices

 [-1-1j   2-1j   2+1j; -1+1j  -2     -2-1j; -2      1      1-1j] -1-2j -1+2j -2j  5 2 1
 [ 1-1j   1+1j   1+1j;  1+1j   1-1j   1+1j;  1-1j   1-1j  -1   ]  3 -2j -2        3 2 2
 [ 1-1j  -1+1j   1-1j;  1-1j   1-1j  -1-1j;  1+1j  -1+1j  -1   ]  3 -2j -2        3 2 2
 [ 1        1j   1   ;    1j    -1j    -1j;  1       -1j   1   ]  2 -2j j         2 2 1
 [-1+1j  -1+1j   1-1j;  1+1j    -1j   1-1j; -1+1j  -1-1j  -1+1j]  3j -2 -2j       3 2 2
 [   1j    -1j     1j;    1j  -1       -1j;    1j     1j     1j]  2j -2 1         2 2 1
 [-1        1j  -1   ;    1j     1j    -1j; -1       -1j  -1   ] -2 2j -j         2 2 1
 [  -1j  -1     -1   ;  1      1     -1   ;  1     -1      1   ] -2j 2 j          2 2 1
 [  -1j     1j     1j;   -1j   1       -1j;    1j     1j    -1j] -2j 2 -1         2 2 1
 [  -1j     1j    -1j;    1j  -1     -1   ;   -1j  -1     -1   ] -2j -2 j         2 2 1
 [ 1       -1j     1j;   -1j     1j  -1   ;   -1j   1      1   ]  2 2j -j         2 2 1
 [ 1+1j  -1+1j  -1+1j; -1+1j   1+1j  -1+1j;  1+1j   1+1j    -1j]  3j -2j 2        3 2 2
 [-1      1     -1   ;  1     -1     -1   ;  1      1        1j] -2 2j -j         2 2 1
 [ 1+1j  -1-1j   1+1j;  1-1j    -1j  -1-1j; -1-1j   1-1j   1+1j]  3j 2 -2j        3 2 2
 [   1j     1j  -1   ;    1j    -1j  -1   ;  1      1       -1j]  2j -2j -j       2 2 1
 [   1j    -1j     1j;   -1j     1j     1j;    1j     1j    -1j] -2j 2j j         2 2 1
 [-1-1j   1-1j  -1-1j;  1-1j  -1-1j   1+1j; -1+1j   1-1j   1   ] -3 2 -2j         3 2 2
 [-1-1j  -1-1j  -1+1j; -1+1j   1     -1+1j; -1+1j  -1-1j  -1-1j] -3 -2j 2         3 2 2
 [  -1j  -1     -1   ;  1       -1j     1j; -1       -1j  -1   ] -2 -2j 1         2 2 1
 [  -1j     1j    -1j;    1j     1j     1j;   -1j     1j     1j] -2j 2j j         2 2 1
 [-1     -1        1j;  1       -1j     1j;   -1j     1j  -1   ] -2 -2j j         2 2 1
 [   1j  -1+1j  -2   ;    2j     2j  -1-1j; -2-2j  -2       -1j] -2-j 2+j 2j      5 2 1

The inverse of the last is particularily nice:

   0.12-0.16j   0.3+0.1j  -0.08+0.44j
  -0.4 +0.2 j   0  -0.5j   0.1 -0.3j
  -0.32-0.24j   0.2+0.4j  -0.12+0.16j

The one matrix that is almost all real integers also is nice:

  -1  1 -1
   1 -1 -1
   1  1  j

with the inverse

1    / -1+j   1+j  2 \
- * |   1+j  -1+j  2  |
4    \ -2    -1    0 /

with eigenvalues -2, 2j and -j and singular values 2, 2, 1.