6.14 Tensor Type One One Classification

Tensor Type One One Classification is the category into which every tensor whose contravariant order equals one and whose covariant order equals one is placed, this category coinciding exactly with the linear operators mapping the underlying vector space to itself. A tensor belonging to this classification accepts a single one-form argument and a single vector argument together, returns a scalar through a bilinear pairing of the two, carries exactly one upper index and one lower index in its component representation, and transforms under a change of coordinates by exactly one factor of the direct Jacobian matrix together with exactly one factor of the inverse Jacobian matrix.

Criteria for Membership in the Classification

Exactly One Upper Index and One Lower Index

Membership in the type one-one classification requires the contravariant order to equal precisely one and the covariant order to equal precisely one simultaneously; a tensor with two upper indices and no lower index, or with one upper index and two lower indices, belongs to a different classification even though each carries at least one index of the relevant kind. The classification is defined by the exact type pair, not merely by the presence of some mixture of variances.

Identity With Linear Operators on the Vector Space

Every element of the type one-one classification corresponds, by construction, to a linear operator on the underlying vector space, since fixing the vector argument in the lower slot and leaving the upper slot open, or equivalently contracting the lower index against a supplied vector's components, produces a new vector, exhibiting the tensor as a machine that turns vectors into vectors through a linear rule. This classification therefore reproduces the space of linear operators as one particular layer within the larger hierarchy of tensor types.

Component Structure and Transformation

Two Indices, n Squared Components

A type one-one tensor's components form an array indexed by one upper index and one lower index, each ranging across all basis directions of a space of dimension n, giving exactly n squared independent numbers needed to specify the tensor once a basis has been fixed, matching precisely the number of entries in an n by n matrix representing the same operator.

One Direct Factor and One Inverse Factor

The transformation law for a type one-one tensor consists of exactly one factor of the direct Jacobian matrix, contributed by the upper index, and exactly one factor of the inverse Jacobian matrix, contributed by the lower index, both contracted against the original components.

Position of the Classification Within the Hierarchy

The Simplest Mixed Type

Type one-one is the simplest classification in the hierarchy possessing both a positive contravariant order and a positive covariant order simultaneously, sitting one step beyond both type one-zero, reached by adding a covariant index, and type zero-one, reached by adding a contravariant index. It is the smallest member of the broader mixed type category.

The Kronecker Delta as Its Distinguished Element

Among all type one-one tensors, the Kronecker delta occupies a distinguished position, being the unique tensor of this type that reduces, in every basis simultaneously, to the identity operator, equal to one whenever its upper and lower index coincide and zero otherwise. Its transformation law, one direct factor and one inverse factor, composes to leave it invariant under any change of coordinates, a property not shared by a generic type one-one tensor.

Closed Under Composition Through Contraction

The collection of all type one-one tensors is closed under an operation of composition formed by contracting the upper index of one tensor against the lower index of another, producing a new type one-one tensor whose action reproduces the successive application of the two original operators. This composition endows the classification with an algebraic structure beyond the simple vector space structure shared by every tensor type, since composition of linear operators is generally noncommutative and does not reduce to ordinary addition or scalar multiplication.