Scalar

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

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Generally, the setting is that of a (ground) field $ F $( more generally, a ring $ R $) and a vector space $ V $( of functions, vectors, matrices, tensors, etc.) over it (more generally, a module $ M $). The elements of $ F $( respectively, $ R $) are called scalars. If $ V $( respectively, $ M $) is an algebra with unit element $ e $, the elements $ \lambda e $, $ \lambda $ in $ F $( respectively, $ R $) are also called scalars. For example, one sometimes speaks of the ( $ n \times n $)- matrices $ \mathop{\rm diag} ( \lambda \dots \lambda ) $ as scalar matrices. The scalar multiples of an element $ v \in V $( respectively, $ m \in M $) are the elements $ \lambda v $, $ \lambda \in F $( respectively, $ \lambda m $, $ \lambda \in R $).

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 $ M $ is a function on $ M $; that is, a scalar field, or field of scalars, is a tensor field (cf. Tensor bundle) of rank $ ( 0, 0) $. These are the scalars in the algebra of tensor fields on $ M $ over the ring of functions on $ M $.

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

Given a left module $ M $ over a ring $ R $ and an $ R $- algebra $ S $, one forms the tensor product $ S \otimes _ {R} M $. This is a module over $ S $. The module $ S \otimes _ {R} M $ is said to be obtained from $ M $ by extension of scalars.

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