16.8.4 Tensor Alternating Bilinear Diagonal Vanishing

Tensor Alternating Bilinear Diagonal Vanishing is the specific fact, isolated at the level of the matrix representation, that every diagonal entry of the skew-symmetric matrix representing an alternating bilinear form is zero, together with its consequences for the associated quadratic form, trace, and diagonal-dependent computations.

The Diagonal Vanishing Fact

Statement in Matrix Terms

For the matrix M representing an alternating bilinear form B relative to any basis:

This holds in every basis simultaneously, not just a specially chosen one, since the diagonal vanishing follows from Mᵀ = −M applied entrywise to the diagonal, where transposition leaves diagonal positions fixed.

Derivation from the Skew-Symmetric Condition

Because Mᵀ = −M means M_{ji} = −M_{ij} for all i,j, setting j = i gives M_{ii} = −M_{ii}, and sign cancellation (valid whenever 2 is invertible) forces M_{ii} = 0 directly from the matrix condition alone, without needing to invoke B as a multilinear map at all.

Consequence for the Associated Quadratic Form

The Quadratic Form Is Identically Zero

Every bilinear form B has an associated quadratic form Q(v) = B(v,v); for an alternating B, diagonal vanishing at the component level combines with bilinearity to force Q to vanish identically on all of V, not merely on basis vectors:

This is a stronger statement than diagonal vanishing alone, since it requires the off-diagonal contributions v_iM_{ij}v_j and v_jM_{ji}v_i to cancel pairwise, which they do precisely because M_{ji} = −M_{ij}.

Contrast with Symmetric Bilinear Forms

For a symmetric bilinear form, by contrast, the diagonal entries need not vanish, and the associated quadratic form is generally nonzero and carries essential information (as in an inner product, where Q(v) = ‖v‖²); diagonal vanishing is therefore a distinguishing feature that separates the alternating case sharply from the symmetric case at the level of the associated quadratic form.

Consequence for the Trace

Trace of the Matrix Representation

Since the trace of a matrix is the sum of its diagonal entries, diagonal vanishing immediately gives:

for every alternating bilinear form's matrix representation, in every basis, since trace is basis-independent.

Trace Vanishing as a Necessary but Not Sufficient Condition

While every skew-symmetric matrix has trace zero, not every trace-zero matrix is skew-symmetric; diagonal vanishing (and hence trace vanishing) is a necessary consequence of the alternating property, but recovering the full skew-symmetric condition additionally requires the off-diagonal relation M_{ij} = −M_{ji} to hold as well.

Diagonal Vanishing Under Basis Change

Preserved, Not Merely Coincidental

Diagonal vanishing survives any congruence transformation M' = AᵀMA, since the derivation above depends only on M being skew-symmetric, a property preserved under congruence; this confirms diagonal vanishing is an intrinsic feature of B, appearing in every possible matrix representation regardless of the basis chosen to construct it.

Diagram of the Diagonal Vanishing Pattern