Scalar

A quantity all values of which can be expressed by one (real) number. More generally, a scalar is an element of some field.

Comments

Generally, the setting is that of a (ground) field (more generally, a ring ) and a vector space (of functions, vectors, matrices, tensors, etc.) over it (more generally, a module ). The elements of (respectively, ) are called scalars. If (respectively, ) is an algebra with unit element , the elements , in (respectively, ) are also called scalars. For example, one sometimes speaks of the ()-matrices as scalar matrices. The scalar multiples of an element (respectively, ) are the elements , (respectively, , ).

The term "scalar" comes from the original meaning as a quantity which can be completely specified by one (real) number.

A scalar field on a manifold is a function on ; that is, a scalar field, or field of scalars, is a tensor field (cf. Tensor bundle) of rank . These are the scalars in the algebra of tensor fields on over the ring of functions on .

A scalar operator on, say, a complex Banach space is a scalar multiple of the identity operator.

Given a left module over a ring and an -algebra , one forms the tensor product . This is a module over . The module is said to be obtained from by extension of scalars.