9.15.2 Tensor Dual Basis Kronecker Pairing

The Kronecker pairing connects dual bases in tensor algebras, revealing duality through bilinear forms in multilinear algebra.

Tensor Dual Basis Kronecker Pairing is the specific formulation of the duality condition between a basis and its dual basis expressed using the Kronecker delta, stating that the pairing of a dual basis covector with a basis vector equals one when their indices coincide and equals zero otherwise. It is the exact algebraic statement, written with a single symbol, that defines what it means for a basis and a set of covectors to be dual to one another.

The Kronecker Delta Statement

Formal Definition

The pairing is written as the application of a dual basis covector to a basis vector, set equal to the Kronecker delta symbol, which itself is defined to take the value one when its two indices are equal and zero when they differ.

e^{i} (e_{j}) = δ_{j}^{i}

Meaning of the Two Cases

When the upper index i and the lower index j are equal, the Kronecker delta equals one, expressing that a dual basis covector applied to its own matching basis vector returns unity. When i and j differ, the Kronecker delta equals zero, expressing that a dual basis covector applied to any other basis vector returns nothing.

δ_{j}^{i} = \{\begin{matrix} 1 & if i = j \\ 0 & if i ≠ j \end{matrix}

Role of the Kronecker Pairing

Complete Specification of the Dual Basis

The Kronecker pairing condition, applied across every combination of indices, uniquely specifies the dual basis once the primary basis is fixed. No other set of covectors besides the one satisfying this condition for every index pair qualifies as the dual basis of the given primary basis.

Basis for Coordinate Extraction

The Kronecker pairing is what makes it possible to extract a single coordinate of a vector by applying the matching dual basis covector, since expanding the vector in the primary basis and applying the dual basis covector leaves only the one term whose Kronecker delta factor equals one.

e^{i} (v^{j} e_{j}) = v^{j} δ_{j}^{i} = v^{i}

Kronecker Pairing and Basis Change

Preservation Under Correct Transformation

When the primary basis and dual basis are transformed together according to the standard basis change rule, the Kronecker pairing condition continues to hold between the new dual basis covectors and the new basis vectors, confirming that the transformation was carried out correctly.

Identifying an Incorrect Dual Basis

If a proposed set of covectors fails to satisfy the Kronecker pairing condition against a given basis, that set does not constitute the dual basis of that basis, regardless of how it was constructed, since the Kronecker pairing is the defining test of duality.

Broader Use of the Kronecker Delta in This Context

Appearing in Index Contractions

The same Kronecker delta symbol that expresses the pairing condition also appears whenever a contraction between an index and its dual counterpart produces a delta rather than a general tensor, marking places in a calculation where the duality pairing has been invoked directly.

Identity Behavior in Mixed Index Expressions

Because the Kronecker delta equals one only on matching indices and zero otherwise, multiplying any indexed quantity by a Kronecker delta and summing over the shared index simply replaces that index with the other one, a substitution property that follows directly from the Kronecker pairing definition.