458
E Quaternions
The matrix elements A ij are scalars and commute with one another. Once the sign
in front of a given term is determined, the actual order in which they are written is
not important.
The same definition of the determinant applies to the matrix ¯
Q N if it is viewed
as a 2N×2N matrix of scalar matrix elements, and we obtain a scalar number
for the value of Det[ ¯
Q N ]. However, when working with quaternions, it is useful
to introduce a new type of determinant, Qdet[ ¯
Q N ], which we call the quaternion
determinant. The quaternion determinant is an incomplete determinant of the matrix
¯
Q N . Its value is a 2×2 matrix. The determinant of this 2×2 matrix must be equal to
the determinant of ¯
Q N . That is, we must have Det[Qdet[ ¯
Q N ]] = Det[ ¯
Q N ].
The quaternion determinant is particularly useful when dealing with self-dual
quaternions. If ¯
Q N is self-dual, then the determinant of ¯
Q N and the quaternion
determinant of ¯
Q N are related. If
Det[ ¯
Q N ] = A, then Qdet[ ¯
Q N ] =
√
A 0
0
√
A
.
(E.16)
The Hamiltonian for a time reversal invariant system is self-dual, and this is the
case for the Gaussian orthogonal and Gaussian symplectic ensembles as well as the
circular orthogonal and circular symplectic ensembles.
Self-dual quaternions ¯
Q N have the property that the sum of a closed cycle and
its inverse closed cycle is a diagonal 2×2 matrix,
( ¯
Q N ) j 1 ,j 2 ( ¯
Q N ) j 2 ,j 3 ×. . .×( ¯
Q N ) j n ,j 1
+( ¯
Q N ) j 1 ,j n ×. . .×( ¯
Q N ) j 3 ,j 2 ( ¯
Q N ) j 2 ,j 1 = C n ¯
1 2 ,
(E.17)
where C n is a constant and ¯
1 2 is a 2×2 unit matrix. We will keep this property of
self-dual quaternions in mind when defining the quaternion determinant of a selfdual quaternion.
Let us now define the quaternion determinant of a self-dual quaternion ¯
Q N . It is
Qdet[ ¯
Q N ] =
P
j 1 ,j 2 ,...,j N ( ¯
q 1,j 1 · ¯
q 2,j 2 ·. . .· ¯
q N,j N ) pcc ,
(E.18)
where (. . .) pcc indicates that the matrix elements must be arranged in products of
closed cycles (pcc) or the inverse closed cycles that naturally appear in various
terms. For example, if one can arrange the closed cycle ¯
q 3,4 · ¯
q 4,8 · ¯
q 8,12 · ¯
q 12,3 in one
term, there will be another term in which the inverse cycle, ¯
q 3,12 · ¯
q 12,8 · ¯
q 8,4 · ¯
q 4,3
can be formed. Because the 2×2 quaternions do not commute, we find that
¯
q 3,4 · ¯
q 4,8 · ¯
q 8,12 · ¯
q 12,3 = ¯
q 4,8 · ¯
q 3,4 · ¯
q 8,12 · ¯
q 12,3 , so the ordering of the 2×2 quaternion
matrix elements is critical. In order to make this clear, below we give two examples
of quaternion determinants.
E Quaternions
The matrix elements A ij are scalars and commute with one another. Once the sign
in front of a given term is determined, the actual order in which they are written is
not important.
The same definition of the determinant applies to the matrix ¯
Q N if it is viewed
as a 2N×2N matrix of scalar matrix elements, and we obtain a scalar number
for the value of Det[ ¯
Q N ]. However, when working with quaternions, it is useful
to introduce a new type of determinant, Qdet[ ¯
Q N ], which we call the quaternion
determinant. The quaternion determinant is an incomplete determinant of the matrix
¯
Q N . Its value is a 2×2 matrix. The determinant of this 2×2 matrix must be equal to
the determinant of ¯
Q N . That is, we must have Det[Qdet[ ¯
Q N ]] = Det[ ¯
Q N ].
The quaternion determinant is particularly useful when dealing with self-dual
quaternions. If ¯
Q N is self-dual, then the determinant of ¯
Q N and the quaternion
determinant of ¯
Q N are related. If
Det[ ¯
Q N ] = A, then Qdet[ ¯
Q N ] =
√
A 0
0
√
A
.
(E.16)
The Hamiltonian for a time reversal invariant system is self-dual, and this is the
case for the Gaussian orthogonal and Gaussian symplectic ensembles as well as the
circular orthogonal and circular symplectic ensembles.
Self-dual quaternions ¯
Q N have the property that the sum of a closed cycle and
its inverse closed cycle is a diagonal 2×2 matrix,
( ¯
Q N ) j 1 ,j 2 ( ¯
Q N ) j 2 ,j 3 ×. . .×( ¯
Q N ) j n ,j 1
+( ¯
Q N ) j 1 ,j n ×. . .×( ¯
Q N ) j 3 ,j 2 ( ¯
Q N ) j 2 ,j 1 = C n ¯
1 2 ,
(E.17)
where C n is a constant and ¯
1 2 is a 2×2 unit matrix. We will keep this property of
self-dual quaternions in mind when defining the quaternion determinant of a selfdual quaternion.
Let us now define the quaternion determinant of a self-dual quaternion ¯
Q N . It is
Qdet[ ¯
Q N ] =
P
j 1 ,j 2 ,...,j N ( ¯
q 1,j 1 · ¯
q 2,j 2 ·. . .· ¯
q N,j N ) pcc ,
(E.18)
where (. . .) pcc indicates that the matrix elements must be arranged in products of
closed cycles (pcc) or the inverse closed cycles that naturally appear in various
terms. For example, if one can arrange the closed cycle ¯
q 3,4 · ¯
q 4,8 · ¯
q 8,12 · ¯
q 12,3 in one
term, there will be another term in which the inverse cycle, ¯
q 3,12 · ¯
q 12,8 · ¯
q 8,4 · ¯
q 4,3
can be formed. Because the 2×2 quaternions do not commute, we find that
¯
q 3,4 · ¯
q 4,8 · ¯
q 8,12 · ¯
q 12,3 = ¯
q 4,8 · ¯
q 3,4 · ¯
q 8,12 · ¯
q 12,3 , so the ordering of the 2×2 quaternion
matrix elements is critical. In order to make this clear, below we give two examples
of quaternion determinants.