16.14.4 Tensor Determinant Permutation Expansion

The Tensor Determinant Permutation Expansion connects multilinear algebra with combinatorics through signed sums over permutations.

Tensor Determinant Permutation Expansion is the explicit representation of the determinant of an n by n matrix as a signed sum over all n! permutations of its column indices, each term being the product of matrix entries selected according to that permutation and weighted by the permutation's sign. It is the classical Leibniz formula, and it serves as the direct computational realization of the abstract characterization of the determinant as the unique normalized alternating multilinear form.

Statement of the Expansion

The Leibniz Formula

For an n by n matrix A with entries aᵢⱼ, the determinant is given by:

det (A) = \sum_{σ \in S_{n}} sgn (σ) \prod_{i = 1}^{n} a_{i σ (i)}

where the sum ranges over every permutation σ in the symmetric group Sₙ, sgn(σ) is +1 for an even permutation and −1 for an odd permutation, and each product selects one entry from every row and every column according to the permutation.

Interpretation of Each Term

Each term in the sum corresponds to a choice of one entry from each row such that no two chosen entries share a column, which is precisely what a permutation σ encodes: σ(i) specifies the column selected for row i. The sign attached to each term reflects the parity of the number of transpositions needed to realize σ, which in turn reflects how the alternating property of the determinant reorders arguments.

Derivation From Multilinearity and Alternation

Expanding via Standard Basis Vectors

Writing each column vᵢ of the matrix as a linear combination of standard basis vectors and expanding the determinant by multilinearity in each of the n column arguments produces a sum over all possible ways of assigning a basis vector index to each column, giving nⁿ terms before simplification.

Eliminating Repeated Indices

Among these nⁿ terms, any term in which two columns are assigned the same basis vector index contains a repeated argument and vanishes immediately by the alternating argument rule. Only the terms in which the assignment of indices to columns forms a bijection, that is, a permutation of {1, ..., n}, survive this elimination, reducing the sum from nⁿ terms to exactly n! surviving terms.

Collecting Sign Factors

Each surviving term corresponds to det(e_{σ(1)}, ..., e_{σ(n)}) multiplied by the appropriate product of matrix entries. Since permuting the basis vectors reduces to a sequence of transpositions, and each transposition multiplies the determinant by −1 according to the alternating rule, det(e_{σ(1)}, ..., e_{σ(n)}) equals sgn(σ) times det(e₁, ..., eₙ), which equals sgn(σ) by the normalization condition. This yields exactly the Leibniz formula.

Consequences of the Expansion

Recovering Small Determinant Formulas

For a 2 by 2 matrix, the permutation expansion has only 2! = 2 terms, corresponding to the identity permutation and the single transposition, giving the familiar formula a₁₁a₂₂ − a₁₂a₂₁. For a 3 by 3 matrix, the expansion has 3! = 6 terms, three with positive sign and three with negative sign, reproducing the rule of Sarrus used in elementary linear algebra courses.

Growth of Computational Complexity

Because the number of terms in the permutation expansion grows as n!, direct evaluation of the determinant via this formula becomes computationally impractical for large n, which is why cofactor expansion and row reduction techniques, both consequences of the same underlying multilinear and alternating properties, are preferred in practice despite being derived from the identical formula.

Uniqueness of the Formula

Since the Leibniz formula is derived purely from multilinearity, the alternating argument rule, and the normalization condition, it is the unique formula satisfying all three defining properties simultaneously. Any other function claiming to be a determinant on n by n matrices satisfying these three properties must coincide exactly with this permutation sum.

Connection to the Exterior Algebra

Sign of a Permutation as a Wedge Product Reordering

The sign attached to each permutation in the expansion is exactly the sign that arises when reordering a wedge product of basis vectors into standard increasing order. This ties the combinatorial sign convention used in the Leibniz formula directly to the antisymmetry mechanism operating within the top exterior power Λⁿ(V).

Significance of the Expansion

The permutation expansion provides the explicit, closed-form definition of the determinant from which every other determinant formula and identity can, in principle, be derived. It makes visible the precise combinatorial mechanism, the interplay of permutations and their signs, that underlies the abstract characterization of the determinant as the unique normalized alternating multilinear form on an n-dimensional vector space.