(where the element in row and column is , not ). Thus an arbitrary element of is of the form
(sometimes denoted by or by ) and the product of two elements and (order matters) is where
The elements of are known as Hamiltonian quaternions.
Clearly the subspaces of generated by and by are subalgebras isomorphic to and respectively. is customarily identified with the corresponding subalgebra of . (We shall see in a moment that there are other and less obvious embeddings of in .) The real numbers commute with all the elements of , and we have
for and .
If and , then
with equality only if
which means that qualifies as a normed algebra when we give it the norm .
Because the norm of any nonzero quaternion is real and nonzero, we have
which shows that any nonzero quaternion has an inverse:
Other embeddings of into
One can use any non-zero to define an embedding of into . If is a natural embedding of into , then the embedding:
is also an embedding into . Because is an associative algebra, it is obvious that:
and with the distributive laws, it is easy to check that
Rotations in 3-space
Let us write
An arbitrary element of can be expressed , for some real numbers such that . The permutation of thus gives rise to a permutation of the real sphere. It turns out that that permutation is a rotation. Its axis is the line through and , and the angle through which it rotates the sphere is . If rotations and correspond to quaternions and respectively, then clearly the permutation corresponds to the composite rotation . Thus this mapping of onto is a group homomorphism. Its kernel is the subset of , and thus it comprises a double cover of . The kernel has a geometric interpretation as well: two unit vectors in opposite directions determine the same axis of rotation.
|Date of creation||2013-03-22 12:35:42|
|Last modified on||2013-03-22 12:35:42|
|Last modified by||mathcam (2727)|