Definition:Matrix/Underlying Structure

Definition
Let $\mathbf A$ be a matrix over a set $S$.

The set $S$ can be referred to as the underlying set of $\mathbf A$.

In the context of matrices, however, it is usual for $S$ itself to be the underlying set of an algebraic structure in its own right.

If this is the case, then the structure $\struct {S, \circ_1, \circ_2, \ldots, \circ_n}$ (which may also be an ordered structure) can be referred to as the underlying structure of $\mathbf A$.

Matrices themselves, when over an algebraic structure, may themselves have operations defined on them which are induced by the operations of the structures over which they are formed.

However, because the concept of matrices was originally developed for use over the standard number systems ($\Z$, $\R$ and so on), the language used to define their operations (that is "addition", "multiplication", etc.) tends to relate directly to such operations on those underlying number systems.

The concept of the matrix can be extended to be used over more general algebraic structures than these, and it needs to be borne in mind that although the matrix operations as standardly defined may bear the names of those familiar "numerical" operations, those of the underlying structure may not necessarily be so.