12.15.4 Tensor Direct Sum Injection Projection Pair

The Tensor Direct Sum Injection Projection Pair explores how tensor spaces combine and decompose through injective and projective mappings in algebra.

Tensor Direct Sum Injection Projection Pair is the matched pair of canonical maps (ιₖ, πₖ) associated with each summand Vₖ of a direct sum V = V₁ ⊕ ... ⊕ Vₙ, consisting of the inclusion ιₖ : Vₖ → V that embeds the summand into the whole and the projection πₖ : V → Vₖ that recovers it, satisfying together the retraction identity πₖ ∘ ιₖ = id_Vₖ and the annihilation identities πⱼ ∘ ιₖ = 0 for j ≠ k. This pair is the precise algebraic packaging of the informal idea that a direct sum lets each summand be "put in" and "taken back out" without interference from the other summands.

The Two Maps Individually

The Injection Component

ι_{k} (v_{k}) = (0, \dots, v_{k}, \dots, 0)

The injection ιₖ places vₖ in the k-th coordinate and zero everywhere else. It is precisely an instance of the tensor inclusion operation, satisfying the embedding map role: it is linear, and it is injective since two distinct elements vₖ ≠ vₖ′ produce tuples differing in the k-th coordinate.

The Projection Component

π_{k} (v_{1}, \dots, v_{n}) = v_{k}

The projection πₖ reads off the k-th coordinate and discards the rest. It is an instance of the tensor projection operation, satisfying linearity directly from the coordinate-extraction formula and idempotency, since extracting the k-th coordinate of an already-extracted-and-reinserted tuple returns the same value.

The Defining Identities of the Pair

Retraction: Recovering What Was Injected

π_{k} (ι_{k} (v_{k})) = v_{k}

Injecting vₖ into slot k and then projecting back out of slot k returns exactly the original element. This is the direct sum's realization of the general retraction identity π ∘ ι = id studied for arbitrary paired inclusions and projections, here holding automatically and without any auxiliary hypothesis, purely because of how ιₖ and πₖ are built from the same coordinate k.

Annihilation: Cross-Slot Interference Vanishes

π_{j} (ι_{k} (v_{k})) = 0 ​ ​ for ​ ​ j \neq k

Injecting into slot k and then projecting out of a different slot j always yields zero, since injection places zero in every slot other than k. This annihilation property has no counterpart requirement in the general theory of a single inclusion-projection pair; it only becomes meaningful once several such pairs coexist within the same direct sum, and it is what guarantees the different summands genuinely do not interfere with one another.

Completeness: Recombination Yields the Identity

\sum_{k} ι_{k} \circ π_{k} = {id}_{V}

Summing the composite ιₖ ∘ πₖ over every index k reproduces the identity map on the whole direct sum, since each term reinserts exactly the k-th coordinate into its own slot and zero elsewhere, and summing across all k reconstructs every coordinate simultaneously. This completeness identity is what confirms the family of injection-projection pairs accounts for the entire structure of V, with nothing left unaccounted for.

Diagram of a Single Injection-Projection Pair

The Pair as a Complete System

Determining the Direct Sum from Its Pairs

Given a space V and a family of injection-projection pairs (ιₖ, πₖ) satisfying retraction, annihilation, and completeness, V is canonically isomorphic to the direct sum V₁ ⊕ ... ⊕ Vₙ, where Vₖ is identified with the image ιₖ(Vₖ). This shows the injection-projection pair structure is not merely a convenient byproduct of an already-constructed direct sum but a complete, alternative characterization of what it means for a space to decompose as a direct sum in the first place.

Matrix Form of the Pair

Relative to a basis of V adapted to the direct sum decomposition, ιₖ is represented by a matrix consisting of an identity block in rows corresponding to Vₖ and zero elsewhere, while πₖ is represented by the transpose pattern: an identity block in the columns corresponding to Vₖ and zero elsewhere. The retraction identity πₖιₖ = id corresponds to this identity block multiplying itself back to the identity, while the annihilation identity corresponds to mismatched blocks multiplying to a zero matrix.

Relation to General Inclusion and Projection Theory

Special Case with Extra Guarantees

Every individual pair (ιₖ, πₖ) satisfies the general embedding map role and linear map role required of inclusions and projections studied independently, but the direct sum context supplies two additional guarantees — mutual annihilation across different indices and joint completeness summing to the identity — that are not automatic for an arbitrary, unrelated inclusion and projection chosen without reference to a common direct sum structure.

Foundation for Decompose-Compute-Reassemble Workflows

The injection-projection pair is the algebraic tool underlying the strategy of decomposing a complex tensor computation into independent pieces handled one summand at a time, then reassembling the pieces via the completeness identity, with retraction guaranteeing no piece is corrupted in the round trip and annihilation guaranteeing no piece leaks into another during reassembly.