16.18.4 Tensor Alternating Subspace Invariance

Tensor Alternating Subspace Invariance refers to the property of alternating tensors remaining unchanged under specific linear transformations within their subspace.

Tensor Alternating Subspace Invariance is the property that when a linear map preserves a given subspace of a vector space, the induced action on exterior powers correspondingly preserves the exterior power of that subspace as a subspace of the exterior power of the whole space, ensuring that invariant subspace structure at the vector level lifts consistently and predictably to the level of alternating tensors. It captures how the subspace lattice of a vector space interacts coherently with the exterior algebra construction under linear transformations.

Statement of the Invariance

Subspace-Preserving Maps

Suppose T is a linear map from V to itself, and U is a subspace of V satisfying T(U) ⊆ U, meaning U is invariant under T. Alternating subspace invariance asserts that the induced exterior power map Λᵏ(T) similarly satisfies:

Λ^{k} (T) (Λ^{k} (U)) \subseteq Λ^{k} (U)

where Λᵏ(U) is regarded as the subspace of Λᵏ(V) spanned by wedge products of k vectors all drawn from U.

Why This Follows Directly

This invariance follows immediately from the definition of the induced map: for any u₁, ..., uₖ in U, applying Λᵏ(T) sends u₁ ∧ ... ∧ uₖ to Tu₁ ∧ ... ∧ Tuₖ, and since each Tuᵢ remains in U by the invariance of U under T, the resulting wedge product is again an element of Λᵏ(U). Because Λᵏ(U) is spanned by exactly such simple wedge products, this confirms the containment holds for the entire subspace, not merely for individual simple wedge products.

Consequences for Structure

Restriction of the Induced Map

Alternating subspace invariance allows the induced map Λᵏ(T) to be meaningfully restricted to a map on Λᵏ(U) alone, denoted Λᵏ(T)|_U, which coincides with the exterior power of the restricted map T|_U. This restriction is essential when analyzing the behavior of T on an invariant subspace independently of its action on the rest of V.

Compatibility With Direct Sum Decompositions

When V decomposes as a direct sum of T-invariant subspaces U₁ ⊕ U₂, the exterior power Λᵏ(V) decomposes correspondingly into pieces built from wedging vectors within a single summand together with pieces built from wedging vectors across different summands, and alternating subspace invariance ensures that the induced map Λᵏ(T) respects this decomposition in a controlled way, acting on the purely within-summand pieces according to the invariance property established for each Uᵢ individually.

Application to Eigenspaces

Invariance of Eigenspace Exterior Powers

If U is an eigenspace of T corresponding to eigenvalue λ, meaning T acts on U by scalar multiplication by λ, then alternating subspace invariance guarantees that Λᵏ(T) acts on Λᵏ(U) by scalar multiplication by λᵏ, since each of the k vectors being wedged is individually scaled by λ, and the wedge product is multilinear in each factor:

Λ^{k} (T) (u_{1} \land \dots \land u_{k}) = λ^{k} \cdot u_{1} \land \dots \land u_{k}

Relation to Eigenvalues of the Exterior Power Map

This special case illustrates a broader fact: the eigenvalues of Λᵏ(T), when T is diagonalizable with eigenvalues λ₁, ..., λₙ, are exactly the products of every k-element subset of these eigenvalues, since each eigenspace exterior power invariance combines multiplicatively across the k factors of any wedge product drawn from a mixture of eigenspaces, following the same reasoning extended across multiple eigenvalues simultaneously.

Role in Structural Analysis

Simplifying Block-Triangular Computations

When T has a block upper triangular matrix representation relative to a flag of invariant subspaces, alternating subspace invariance allows the exterior power map to be analyzed one invariant piece at a time, since the wedge products entirely contained within an invariant subspace transform independently of vectors outside it, simplifying what would otherwise be an unwieldy computation across the full exterior power.

Significance of the Invariance

Alternating subspace invariance is what allows the rich subspace structure of linear algebra, invariant subspaces, eigenspaces, and flags, to be transported coherently into the exterior algebra setting. It underlies the multiplicative relationship between the eigenvalues of a linear map and the eigenvalues of its induced exterior power maps, and it enables structural decomposition techniques that analyze exterior power behavior one invariant piece at a time rather than requiring direct computation across the entire space at once.