12.13.5 Tensor Projection Result Structure

Tensor Projection Result Structure is the collection of algebraic properties that the output of a tensor projection operation is guaranteed to carry, independent of which particular input tensor produced it — namely that the result always lies in a fixed subspace (the image of the projection), that it is stable under repeated projection, and that it retains whatever tensor rank, symmetry, or variance type is appropriate to the target space rather than the source space. Where the target component describes the output for one specific input, the result structure describes the shared shape and constraints obeyed by every output the projection can ever produce.

The Image as the Ambient Result Structure

Every Result Lies in a Fixed Subspace

Regardless of which tensor t in the source space T is projected, the result π(t) always lies in im(π), a single fixed subspace determined entirely by π and independent of t. This is the most basic component of the result structure: the range of possible outputs is constrained in advance, not discovered anew for each input.

The Image Is Itself Closed Under the Projection

Because π ∘ π = π, every element already inside im(π) is fixed by π: for w ∈ im(π), writing w = π(u) for some u, π(w) = π(π(u)) = π(u) = w. The result structure therefore includes the fact that π acts as the identity on its own image, so results of the projection are never further altered by projecting them again.

Rank and Type Adjustments in the Result

Rank Reduction Under Contraction-Based Projections

When the projection operation is realized as an index contraction against a fixed covector, the result structure includes a definite drop in tensor rank by one for each contracted index, since one covariant or contravariant slot is consumed by the contraction. This rank behavior is a structural fact about the projection itself, holding uniformly across every tensor it is applied to.

Variance Type of the Result

If the source tensor has mixed variance, say p contravariant and q covariant indices, a projection that contracts one covariant index against a fixed vector produces a result with p contravariant and q − 1 covariant indices. The result structure specifies this new variance signature in advance, so that the type of every output tensor is known before any particular input is projected.

Symmetry Preservation or Alteration

A projection onto the symmetric part of a rank-2 tensor always produces a result satisfying π(T)ᵢⱼ = π(T)ⱼᵢ, regardless of whether the original tensor T was symmetric. This symmetry constraint is part of the result structure of that specific projection, distinguishing it from a projection onto the antisymmetric part, whose results always satisfy π(T)ᵢⱼ = −π(T)ⱼᵢ instead.

Diagram of the Result Structure as a Fixed Subspace

Algebraic Constraints Satisfied by Every Result

Membership in a Kernel-Complementary Subspace

Since the source space decomposes as a direct sum of the image and kernel of π, every result of the projection is, by construction, a member of a subspace complementary to ker(π). This means the result structure excludes an entire complementary region of the source space's structure a priori — no output of π can carry any component lying purely in ker(π).

Basis Restriction

If {f₁, ..., fₘ} is a basis for im(π), every result π(t) is expressible as a linear combination of only these m basis vectors, regardless of how many basis vectors were needed to express the original t. The result structure therefore includes a definite reduction in the effective degrees of freedom compared to the source space, from dim(T) down to dim(im(π)) = m.

Result Structure Under Composition of Projections

Nested Results Stay Within the Innermost Image

For a composite π₂ ∘ π₁ of two compatible projections, every result lies in im(π₂ ∘ π₁) ⊆ im(π₂), since applying π₂ last guarantees the final output satisfies whatever result structure π₂ alone imposes. Composing further projections can only maintain or shrink the resulting subspace, never enlarge it beyond what the last-applied projection allows.

Sum-Decomposed Results

When a tensor is processed by a complete family of mutually orthogonal projections π₁, ..., πₙ summing to the identity, the aggregate result structure across the whole family reconstructs the entire source space: each im(πᵢ) contributes a distinct structural piece, and the direct sum of these pieces recovers the full space T, confirming that the family's combined result structure loses no information present in the original tensor.