14.12.2 Tensor Identity Map Tensor Space Action

The Tensor Identity Map acts within tensor space by preserving structure, mapping tensors to themselves through intrinsic algebraic properties.

Tensor Identity Map Tensor Space Action is the description of how the identity map on a tensor product space, or the tensor product of individual identity maps on each factor space, acts on every tensor in that space by leaving it completely unchanged, providing the baseline reference case against which every other operator on the tensor product space is compared.

The Action Itself

Identity on a Single Factor

The identity map on a single factor space sends every vector of that space to itself, without altering its coordinates in any basis.

I_{V} (v) = v

Tensor Product of Identity Maps

When the identity map on the first factor space is combined with the identity map on the second factor space, the resulting combined operator acts on any elementary tensor by leaving both components unchanged, and therefore leaves the elementary tensor itself unchanged.

(I_{V_{1}} \otimes I_{V_{2}}) (u \otimes v) = u \otimes v

Extension to General Tensors

Linear Extension of the Identity Action

By the linear extension of operators defined on elementary tensors, the tensor product of the two identity maps also leaves every general tensor, expressed as a sum of elementary tensors, completely unchanged, since each term in the sum is individually preserved.

(I_{V_{1}} \otimes I_{V_{2}}) = I_{V_{1} \otimes V_{2}}

Coincidence With the Tensor Space Identity

The tensor product of individual identity maps is not merely similar to the identity map on the full tensor product space; it is exactly equal to it, so no distinction remains between the two once the extension to general tensors is carried out.

Diagram of the Identity Action

Every Tensor Maps to Itself

The diagram below depicts a general tensor in the product space returning to itself under the action of the combined identity map, illustrated as a loop rather than a transformation.

Matrix Representation of the Identity Action

Identity Kronecker Product

Relative to any pair of bases fixed for the two factor spaces, the matrix representing the tensor product of identity maps is the Kronecker product of the two identity matrices, which equals the identity matrix of size matching the dimension of the tensor product space.

I_{m_{1}} \otimes I_{m_{2}} = I_{m_{1} m_{2}}

Independence From the Chosen Basis

Because the identity matrix is invariant under conjugation by any invertible change of basis matrix, the matrix representation of the identity action on the tensor product space remains the identity matrix regardless of which bases are chosen for the two factor spaces.

Role as a Neutral Element

Neutral Element Under Composition

Composing the identity action with any other combined operator on the tensor product space leaves that other operator completely unchanged, in either order of composition, so the identity action functions as the neutral element for composition of operators on the tensor product space.

T \circ (I_{V_{1}} \otimes I_{V_{2}}) = T

Partial Identity on a Single Factor

When only one factor operator is the identity while the other is some nontrivial operator, the resulting combined operator acts as the identity on that one factor's component while applying the nontrivial operator to the remaining component, leaving that factor entirely untouched by the transformation.

Extension to Several Factors

Identity Across Many Factors

When the tensor product involves three or more factor spaces, taking the identity map on every individual factor and combining them produces the identity map on the entire multi-factor tensor product space, following the same reasoning as in the two-factor case.

Mixed Identity and Nontrivial Factors

In a combined operator built from several factors where only some of the factor operators are the identity, those factors remain completely unaffected while the remaining, nontrivial factor operators determine the entire effect of the combined operator on the tensor product space.