Linear Algebra 46 | Leibniz Formula for Determinants [dark version]

TL;DR

The determinant can be defined via an n-dimensional volume form that is multilinear, antisymmetric, and normalized on the unit cube.

Briefing Cornell Notes

Briefing

A determinant can be built from an “oriented volume” function by enforcing three rules—multilinearity, antisymmetry, and normalization—and that setup forces the Leibniz formula. The key payoff is a compact expression: the determinant of an n×n matrix becomes a sum over all permutations of products of matrix entries, weighted by whether each permutation is even or odd. This matters because it turns a geometric quantity (signed volume of a parallelepiped) into an explicit algebraic computation.

The construction starts with an n-dimensional volume form: a function that takes n vectors in ℝ^n and returns a real number. It must be linear in each input vector, antisymmetric under swapping any two vectors (swapping two entries flips the sign), and normalized so the unit cube has volume 1. Using linearity, each input vector is expanded in the canonical basis. Writing the vectors as linear combinations of standard unit vectors e_j, the volume form becomes a large sum of terms where each vector contributes one basis index. After applying linearity repeatedly across all n entries, the expression contains n sums—one for each input vector.

Antisymmetry then sharply reduces the work. If two input vectors correspond to the same basis direction (i.e., two indices coincide), antisymmetry forces the volume form to be zero. That means only terms with all indices distinct survive. “All indices distinct” is exactly the structure of a permutation of {1,2,…,n}: a reordering where each number appears once. So the multiple sums collapse into a single sum over all permutations, typically denoted S_n.

The remaining terms evaluate to either +1 or −1. When the basis vectors appear in the standard order, the normalized volume is +1. Swapping two vectors flips the sign, so any permutation’s contribution is determined by how many swaps are needed to reach the standard order. Even numbers of swaps leave the sign unchanged (+1), while odd numbers produce a sign flip (−1). This sign is formalized as the sign of the permutation, often written sgn(σ).

With these pieces, the volume form—and therefore the determinant defined from it—takes the Leibniz form: for a square matrix A, det(A) equals the sum over all permutations σ in S_n of sgn(σ) times the product of entries A_{1,σ(1)} A_{2,σ(2)} … A_{n,σ(n)}. Conceptually, it multiplies one entry from each row and each column exactly once, then adds all such products with the appropriate plus/minus sign. While this gives a clear definition and geometric meaning (signed volume), it also highlights why direct computation becomes impractical for large n, motivating alternative formulas in later material.

Cornell Notes

The determinant can be defined using an n-dimensional “volume form” that takes n vectors in ℝ^n and returns a signed real number. Imposing three constraints—linearity in each argument, antisymmetry under swapping arguments, and normalization on the unit cube—forces the volume form to match the Leibniz formula. Antisymmetry eliminates every term where two basis indices repeat, leaving only terms indexed by permutations σ ∈ S_n. Each surviving term evaluates to sgn(σ), determined by whether σ is even or odd (even swaps give +1, odd swaps give −1). As a result, det(A) becomes a sum over all permutations of sgn(σ) times the product of matrix entries A_{i,σ(i)} across i=1…n.

Why does antisymmetry make most terms in the multilinear expansion vanish?

When the volume form is antisymmetric, swapping two input vectors flips the sign. If two inputs are the same basis direction (two indices coincide), then swapping those two identical inputs changes nothing geometrically but must also flip the sign—so the value must be 0. In the expanded Leibniz-style sum, that means any term with repeated indices contributes zero, leaving only terms where all indices are distinct.

How do “all indices distinct” terms become a sum over permutations S_n?

After expanding each vector into canonical basis vectors e_j, each input contributes an index. The antisymmetry rule kills terms where any two indices match. What remains are assignments of indices where every number 1 through n appears exactly once. That is precisely a permutation σ of {1,…,n}, and the set of all such permutations is denoted S_n.

What determines whether a permutation contributes +1 or −1?

The normalization sets the volume of the standard ordered basis (e_1, e_2, …, e_n) to be 1. Antisymmetry says exchanging two vectors flips the sign. Therefore, any permutation’s sign is determined by the parity of the number of swaps needed to transform the permuted order back to the standard order: an even number of swaps gives +1, an odd number gives −1. This value is the sign of the permutation, sgn(σ).

How does the Leibniz formula translate into matrix entry products?

For an n×n matrix A, each row i corresponds to the i-th input vector. A permutation σ selects exactly one column for each row, so the product uses entries A_{i,σ(i)} for i=1…n. The determinant sums these products over all σ ∈ S_n, weighting each product by sgn(σ). This ensures each row and each column appears exactly once in every term.

Why is the Leibniz formula conceptually useful even though it’s computationally heavy?

The formula gives an explicit definition of det(A) tied to signed (or oriented) volume: it aggregates all ways to pick one entry per row and column, then assigns a plus/minus sign based on permutation parity. But because it requires summing over all n! permutations, direct computation becomes impractical for large n—hence the need for other determinant formulas later.

Review Questions

In the multilinear expansion of the volume form, what condition on indices guarantees a nonzero contribution?
How is sgn(σ) determined from a permutation, and what does it correspond to geometrically?
Write the Leibniz formula for det(A) using σ ∈ S_n and explain what A_{i,σ(i)} represents in each term.

Key Points

1
The determinant can be defined via an n-dimensional volume form that is multilinear, antisymmetric, and normalized on the unit cube.
2
Linearity expands each input vector into canonical basis vectors, producing a sum over basis-index choices.
3
Antisymmetry forces the volume form to be zero whenever two basis indices coincide, eliminating repeated-index terms.
4
Only index assignments with all distinct values survive, which correspond exactly to permutations σ in S_n.
5
Each surviving term evaluates to +1 or −1 depending on whether σ is even or odd, captured by sgn(σ).
6
For an n×n matrix A, det(A)=∑_{σ∈S_n} sgn(σ)∏_{i=1}^n A_{i,σ(i)}.
7
The Leibniz formula directly reflects signed volume but becomes expensive because it sums over n! permutations.

Highlights

Antisymmetry kills every term with repeated indices, collapsing a multi-sum into a permutation sum over S_n.

The sign of a permutation comes from how many swaps are needed to restore the standard basis order: even swaps give +1, odd swaps give −1.

Each Leibniz term multiplies exactly one entry from each row and each column, then the determinant adds all such products with the correct sign.

Topics

Leibniz Formula
Determinants
Volume Form
Permutations
Antisymmetry