14.12.5 Tensor Identity Map Product Preservation

Tensor Identity Map Product Preservation is the property that forming a tensor product of identity maps, taken across any collection of factor spaces, always produces the identity map on the resulting tensor product space, so the property of being an identity map is preserved intact by the tensor product construction rather than being altered or lost.

Statement of the Preservation Property

Preservation for Two Factors

The tensor product of the identity map on the first factor space with the identity map on the second factor space equals the identity map on the tensor product of the two factor spaces.

Reason the Property Holds

The identity map fixes every vector of its own factor space, so a tensor product of identity maps fixes every component of an elementary tensor, and therefore fixes the elementary tensor itself; extending this fact linearly across sums of elementary tensors shows that every tensor in the product space is fixed as well.

Diagram of Preservation

Identity Inputs Produce an Identity Output

The diagram below shows two identity maps, one per factor, combining into a single identity map on the tensor product space, rather than into some other kind of operator.

Preservation Under Repeated Application

Extension to Three or More Factors

If the identity map is taken on every factor space in a tensor product of three or more factors, the resulting combined operator is the identity map on the full multi-factor tensor product space, since the two-factor preservation property can be applied repeatedly, one additional factor at a time.

Consistency With Associativity

Because the tensor product of maps is associative, the preservation property produces the same final identity map on the full tensor product space regardless of how the individual identity maps are grouped together during the repeated combination.

Preservation as a Special Case of Eigenvalue Behavior

Every Vector Is an Eigenvector With Eigenvalue One

Since every vector in each factor space is an eigenvector of the identity map with eigenvalue one, the general rule that eigenvalues of a combined operator multiply across factors gives an eigenvalue of one for every simple tensor formed from these eigenvectors, consistent with the combined operator being the identity.

Full Spectrum Collapsing to a Single Value

Whereas a general combined operator can have many distinct eigenvalues formed as products of the individual factor eigenvalues, a tensor product of identity maps has every eigenvalue collapse to the single value one, reflecting the fact that the resulting operator fixes the entire space.

Distinguishing Preservation From Mere Similarity

Exact Equality, Not Approximate Correspondence

The preservation property states an exact equality between the tensor product of identity maps and the identity map on the product space, not merely that the two operators behave similarly or produce comparable numerical results; every vector in the tensor product space is mapped to itself precisely, with no exceptions.

Basis-Independent Statement

This preservation holds independently of any choice of basis in the factor spaces, since the identity map is characterized purely by its action of fixing every vector, a description that makes no reference to coordinates or a particular basis.

Consequence for the Matrix Representation

Kronecker Product of Identity Matrices

Relative to any fixed bases, the preservation property corresponds to the fact that the Kronecker product of identity matrices of any sizes equals a single identity matrix whose size matches the product of the individual sizes.

Use as a Consistency Check

This matrix-level fact is often used as a straightforward consistency check when verifying a numerical implementation of the Kronecker product, since any implementation that fails to return an identity matrix when given only identity matrices as input reveals an error in the construction.