The real quaternion algebra H is a non-commutative normed 4-dimensional division algebra over the real field R discovered by Hamilton in 1843. Its traditional basis is r = {1, i, j, k}, where multiplication is determined by the rules i2 = j2 = k2 = -1 and [in a cyclical arrangement] i*j = -j*i = k, j*k = -k*j = i, k*i = -i*k = j, like depicted on the stamp below. So an arbitrary element q of H can be written [in a unique way] as a real linear combination q = a*1 + b*i + c*j + d*k. This may be rearranged into q = (a+b*i)*1 + (c+d*i)*j showing that H is a 2-dimensional algebra over the field C of complex numbers with basis s = {1, j}. So next to the real representation [q]r = [a,b,c,d], with respect to r, we have the complex representstion [q]s = [a+b*i, c+d*i]. The algebra H posesses an involution, an anti-automorphism of order 2, viz. q --> q_ = a*1 - b*i - c*j - d*k which extends complex conjugation and is called likewise. In number theory q*q_ = a2 + b2 + c2 + d2 is called the Norm N(q) of q. It follows that q-1 exists ( q-1 = q_/N(q) ) for any nonzero q. The algebra H is a division algebra, indeed. In analysis, the norm |q| of q is given by the square root of N(q), that is, |q| = (a2 + b2 + c2 + d2)1/2, the euclidean norm of [q]r in four-space. One calls a*1=a the scalar part of q and q - a its vectorial part. If qn = bn*i + cn*j + dn*k for n=1,2 (no scalar parts), then q1*q2 has minus the inner product of [q1]r and [q2]r as scalar part and its vectorial part corresponds to the outer [or cross] product of [q1]r and [q2]r. A famous theorem of Frobenius from 1886 states that R and C and H are, up to isomorphism, the only [associative] real division algebras.
, the quaternion skew field H can be represented as the division algebra of all complex matrices of the form
.
In fact the first representation is just the matrix-representation with respect to the basis {1, i} of the R-linear mapping of C into itself given by z --> (a + bi)*z, while the second is the matrix representation with respect to the basis (1, j} of the C-linear mapping of H into itself given by q --> ((a + bi)+(c+di)*j)*q. It is essential here to view H as a right-linear vector space over C, since otherwise the multiplication mapping above is not C-linear, by lack of commutativity. In particular, to recognize the first column, write (a + bi) + (c + di)*j as (a + bi) + j*(c - di).
opposite Cloghane (Dingle peninsula) [23-06-2001] 
