12.13.3 Tensor Projection Linear Map Role

The tensor projection linear map plays a key role in decomposing tensors by projecting them onto subspaces, essential in algebraic structures and tensor analysis.

Tensor Projection Linear Map Role is the requirement that a tensor projection operation act specifically as a linear map between the source and target spaces, meaning that it must respect vector addition and scalar multiplication in the precise sense π(as + bt) = aπ(s) + bπ(t) for all tensors s, t and all scalars a, b drawn from the common base field or ring. This role is what allows a projection to be treated as an object of linear algebra — represented by a matrix relative to chosen bases, decomposed via eigenvalues, and composed with other linear maps — rather than as an arbitrary rule for selecting part of a tensor.

Statement of the Linear Map Role

Additivity

π (s + t) = π (s) + π (t)

For any two tensors s and t in the source space, the projection of their sum must equal the sum of their individual projections. This rules out any projection rule that treats a tensor's decomposition as depending on which other tensors happen to be added to it.

Homogeneity

π (a \cdot t) = a \cdot π (t)

Scaling a tensor by a scalar a before projecting must give the same result as scaling the projected tensor by a afterward. Together with additivity, this establishes π as a linear map in the standard sense used throughout linear algebra.

Why the Linear Map Role Is Necessary

Compatibility with Tensor Coordinates

Because tensors are typically expressed in coordinates relative to a basis, t = Σ cᵢeᵢ, a projection that is not linear could not be computed coordinate-by-coordinate: the value of π(t) would not be recoverable from the values π(eᵢ) and the coefficients cᵢ. Linearity guarantees π(t) = Σ cᵢπ(eᵢ), so the projection is fully determined once its action on a basis is known.

Matrix Representability

π (e_{j}) = \sum_{i} P_{i j} e_{i}

Because π is linear, its action relative to fixed bases of the source and target spaces is captured entirely by a matrix P with entries Pᵢⱼ, obtained by expressing π(eⱼ) in the target basis. This is what allows tensor projections to be implemented as matrix multiplication in computation, and it is unavailable for maps that fail the linear map role.

Consistency with Multilinear Constructions

Tensors are built from multilinear operations, and any operation meant to interact cleanly with tensor products, contractions, or symmetrizations must itself be linear, since those constructions are only guaranteed to behave well under linear maps applied to their inputs. A nonlinear "projection" would fail to commute with the multilinear machinery used to define and manipulate tensors in the first place.

Diagram of the Linear Map Role

Interaction with the Idempotency Role

Linearity Alone Does Not Make a Map a Projection

Linearity is a necessary but not sufficient condition for π to qualify as a projection: a general linear endomorphism need not satisfy π ∘ π = π. The linear map role and the idempotency condition are independent requirements, and it is their conjunction that defines a genuine tensor projection operation rather than an arbitrary linear transformation.

Linearity Enables the Idempotency Check Itself

The idempotency condition π ∘ π = π is a statement about the composite map π ∘ π, and composites of linear maps are only guaranteed to be linear (and hence directly comparable to π as linear maps) because π satisfies the linear map role in the first place. Idempotency is therefore checked and interpreted entirely within the category of linear maps, making linearity the precondition for the idempotency requirement to even be well-posed.

Consequences for Composition and Combination

Linearity Is Preserved Under Composition

If π₁ and π₂ are both linear maps satisfying the linear map role, their composite π₂ ∘ π₁ is automatically linear as well, since composition of linear maps is linear. This ensures that chains of tensor projections remain within the class of linear maps at every stage, preserving matrix representability throughout a sequence of projection operations.

Linear Combinations of Projections

Because projections are linear maps, ordinary linear-algebra operations on maps — such as forming aπ₁ + bπ₂ — are meaningful and again produce linear maps, though such combinations need not themselves be idempotent. The linear map role thus supplies the ambient vector space of linear maps within which more specialized combinations of projections, such as sums of orthogonal projections, are constructed and analyzed.