16.11.1 Tensor Exterior Product Alternating Output
The tensor exterior product produces alternating outputs, critical in multilinear algebra for antisymmetric tensors and wedge products.
Tensor Exterior Product Alternating Output is the specific verification that the result of wedging two alternating forms together is itself alternating, worked out directly from the determinant-based evaluation formula rather than assumed from the general preservation principle, confirming the wedge product's output satisfies the sign-reversal and vanishing-on-repetition properties on its own terms.
Verifying the Sign-Reversal Property Directly
Setting Up the Check
Take a decomposable wedge φ₁ ∧ ... ∧ φₖ and evaluate it on a tuple of vectors using the determinant formula:
Swapping Two Arguments and Tracking the Determinant
Swapping two arguments vₐ and vᵦ swaps the corresponding two columns of the matrix (φᵢ(vⱼ)), and it is a standard determinant fact that swapping any two columns of a matrix negates its determinant:
directly confirming the wedge product's output satisfies the sign-reversal law, since the swap on the left of the evaluation formula corresponds exactly to the column swap on the right.
Verifying the Vanishing-on-Repetition Property Directly
Repeated Argument Produces a Repeated Column
If two arguments vₐ and vᵦ in the tuple are equal, the corresponding two columns of the matrix (φᵢ(vⱼ)) become identical, since every entry φᵢ(vₐ) = φᵢ(vᵦ) for all i.
Determinant with a Repeated Column Is Zero
A standard determinant fact states that a matrix with two identical columns has determinant zero, which directly gives:
confirming vanishing on repetition directly from a fact about determinants, without needing to invoke the general preservation theorem.
Extending the Verification to General Wedge Elements
Beyond Decomposable Products
Not every element of Λᵏ(V*) is a single decomposable wedge φ₁∧...∧φₖ; general elements are linear combinations of such decomposable pieces. Since both sign-reversal and vanishing-on-repetition are properties preserved under linear combination (verified earlier for the general preservation principle), the direct verification carried out here for decomposable wedges extends automatically to every element of Λᵏ(V*), decomposable or not.
Worked Numerical Check
Take φ₁ = e₁*, φ₂ = e₂* in ℝ², and evaluate (φ₁∧φ₂)(e₁,e₂) = det[[1,0],[0,1]] = 1. Swapping arguments gives (φ₁∧φ₂)(e₂,e₁) = det[[0,1],[1,0]] = −1, confirming the sign flip numerically in the simplest nontrivial case.
Why This Direct Verification Matters
Independent Confirmation, Not Circular Reasoning
Since the wedge product is itself often defined via the antisymmetrization operator (already known to produce alternating output by construction), this direct determinant-based verification serves as an independent confirmation using a completely different route — properties of determinants rather than properties of Alt — reinforcing that the alternating nature of the wedge product's output is a robust fact reachable from multiple independent directions rather than an artifact of one particular definitional choice.