Concretely, the structure of as an -algebra is given by a basis of the underlying vector space of , equipped with a multiplication table where is the identity element and otherwise uniquely specified by the equations
and extended by -linearity to all of . The norm on is given by
where given an -linear combination , we define the conjugate . A simple calculation yields
We have canonical left and right module structures on , but as is not commutative, if we want to talk about tensor products of modules, we need to consider bimodules. This also means that ordinary linear algebra as is used over a field is not quite the same when dealing with quaternions. For instance, one needs to distinguish between left and righteigenvalues of matrices in (using the left and right module structures on respectively), and only left eigenvalues relate to the spectrum of the associated linear operator.