12.13 Tensor Projection Operation

Tensor Projection Operation is the operation that maps a tensor in a given source space onto a designated subspace, factor, or index-reduced target space by discarding or contracting the remaining structure, typically realized as a linear map π : T → T′ satisfying an idempotency condition when regarded as an endomorphism, or as a factor-extraction map when the source space is a tensor product of several component spaces. The operation formalizes the many related notions of "taking a piece of a tensor" — restricting to a summand of a direct-sum decomposition, extracting one factor of a tensor product, or contracting a tensor against a fixed covector along one of its index slots — under a single algebraic framework.

Defining Properties

Linearity

A tensor projection is required to be linear over the base field or ring, for all tensors s, t in the source space and all scalars a, b. This is what allows a projection to be described completely by its action on a basis, and what distinguishes an algebraic projection from an arbitrary set-theoretic selection rule.

Idempotency

When the projection is an endomorphism π : T → T of a single space, it must satisfy π ∘ π = π: applying the projection a second time to an already-projected tensor leaves it unchanged. This condition is what earns the operation the name "projection" rather than merely "linear map," and it geometrically encodes that the image of π is a fixed subspace onto which everything is collapsed in one step.

Complementary Decomposition

Idempotency implies that the source space T splits as a direct sum T = im(π) ⊕ ker(π), where im(π) is the subspace the projection lands in and ker(π) is the subspace it annihilates. Every tensor t ∈ T decomposes uniquely as t = π(t) + (t − π(t)), with π(t) ∈ im(π) and t − π(t) ∈ ker(π), since π(t − π(t)) = π(t) − π(π(t)) = π(t) − π(t) = 0.

Realizations of the Projection Operation

Projection onto a Tensor Product Factor

Given a fixed covector or evaluation functional on V₂, a simple tensor v₁ ⊗ v₂ projects to a scalar multiple of v₁, and this rule extends linearly to all of V₁ ⊗ V₂. This realization requires an auxiliary choice — the functional on V₂ — since the tensor product itself does not distinguish a canonical projection onto one factor without such extra data.

Projection via Index Contraction

For a rank-k tensor and a fixed covector φ contracted against one chosen index slot, the projection operation lowers the rank by one: π(T)^{i₁...î_r...i_k} = Σⱼ φⱼ T^{i₁...j...i_k}, where the hat denotes the omitted index. Repeating this operation against several fixed covectors reduces a high-rank tensor step by step to a scalar.

Projection onto a Symmetric or Antisymmetric Part

For a rank-2 tensor T and the index-swap operation σ, the map above projects T onto its symmetric part. It is linear and idempotent — applying it twice reproduces the same symmetric tensor — making it a projection in the precise algebraic sense, with the antisymmetric-part projection ½(T − σ(T)) as its complementary counterpart.

Diagram of a Projection with Its Kernel and Image

Distinction from General Linear Maps

Not Every Linear Map Is a Projection

A linear map L : T → T need not satisfy L ∘ L = L; when it does not, it cannot decompose T into a direct sum of its image and kernel in the manner described above, since L restricted to its own image may act nontrivially rather than as the identity. The projection operation is therefore a strictly stronger notion than general linearity, singled out precisely by the idempotency requirement.

Relation to Eigenvalue Structure

An idempotent linear map has only 0 and 1 as possible eigenvalues, with im(π) being the 1-eigenspace and ker(π) the 0-eigenspace. This gives an equivalent spectral characterization of the tensor projection operation: it is exactly a linear endomorphism diagonalizable with eigenvalues restricted to {0, 1}.

Composition and Combination of Projections

Composing Two Compatible Projections

If π₁ and π₂ are projections on the same space satisfying π₁π₂ = π₂π₁, their composite π₁π₂ is also a projection, with image im(π₁) ∩ im(π₂). This allows successive extraction of finer and finer pieces of a tensor by chaining compatible projection operations.

Sums of Orthogonal Projections

If π₁ and π₂ satisfy π₁π₂ = π₂π₁ = 0, their sum π₁ + π₂ is again a projection, with image im(π₁) ⊕ im(π₂). This construction underlies decompositions of a tensor space into several simultaneous complementary pieces, each recovered by its own projection operation, all summing back to the identity on the source space.