The collected mathematical papers of James Joseph Sylvester...

76.

A WORD ON NONIONS.
[Johns Hopkins University Circulars, I. (1882), pp. 241, 242;
II. (1883), p. 46.]
IN my lectures on Multiple Algebra I showed that if u, v are two matrices
of the second order, and if the determinant of the matrix (z + yv + xz) be
written as
z2 + 2bxz + 2cyz + dx2 + 2exy + fy2
then the necessary and sufficient conditions for the equation vu + uv = 0 are
the following, namely,
b=0, c=0, e=0.
If to these conditions we superadd d = 1, f= 1, and write uv = w, then
2= —1, v2= —1, w2= —1, uv=-v-U=w, vw= —uv-=-, WZt=-u-zW=v;
and 1, u, v, w form a quaternion system. The conditions above stated will be
satisfied if
Det. (z + yv + xu) = z2 + y2 + x2,
which will obviously be the case if
0 1 0 0
-= -1 O, u= 0 O
where 0 = /(- 1). For then
z y+xO
z +yv + txu = +
Hence the matrices
01 101 0 0 00
construed as complex uantities are a linear transformation of the ordinary
construed as complex quantities are a linear transformation of the ordinary