Cauchy–Binet Formula

Concept Overview

When two square matrices A and B of the same order are multiplied, the determinant of their product satisfies det(AB) = det(A)det(B). When A is an m × n rectangular matrix and B is an n × m rectangular matrix (with m ≤ n), neither det(A) nor det(B) is defined individually, yet the product AB is an m × m square matrix with a well-defined determinant.

The Cauchy–Binet formula bridges this gap by expressing det(AB) as the sum over all m-element index subsets S ⊆ {1, ..., n} of the product of the m × m minor of A using columns in S and the m × m minor of B using rows in S.

Mathematical Formulation

For a 2×3 matrix A and a 3×2 matrix B, the ordered 2-element subsets of {1, 2, 3} are S1 = {1, 2}, S2 = {1, 3}, and S3 = {2, 3}. The Cauchy–Binet identity states:

det(AB) = det(A*,{1,2}) det(B{1,2},*) + det(A*,{1,3}) det(B{1,3},*) + det(A*,{2,3}) det(B{2,3},*)

Geometrically, when B = AT, the formula expresses the squared area of the parallelogram spanned by the rows of A in 3D space as the sum of the squared areas of its orthogonal projections onto the three coordinate planes (the Pythagorean theorem for areas).

Worked Example: Cauchy–Binet Expansion

Consider a 2×3 matrix A and a 3×2 matrix B:

A = [ 1  2  0 ]
    [ 0  1  1 ]

B = [ 1  0 ]
    [ 2  1 ]
    [ 0  2 ]

First, we compute the direct product AB:

AB = [ (1)(1) + (2)(2) + (0)(0)    (1)(0) + (2)(1) + (0)(2) ]
     [ (0)(1) + (1)(2) + (1)(0)    (0)(0) + (1)(1) + (1)(2) ]

   = [ 5  2 ]
     [ 2  3 ]

The determinant of this 2×2 matrix is det(AB) = (5)(3) - (2)(2) = 15 - 4 = 11.

The Cauchy–Binet formula expands this determinant as the sum of the products of corresponding maximal minors. For a 2×3 matrix, the maximal minors are 2×2 matrices, chosen by selecting 2 out of the 3 columns (for A) and the same 2 rows (for B). There are exactly 3 such subsets: {1, 2}, {1, 3}, and {2, 3}.

1. Subset {1, 2}:
   Minor of A: det([1, 2; 0, 1]) = 1 - 0 = 1
   Minor of B: det([1, 0; 2, 1]) = 1 - 0 = 1
   Product = (1)(1) = 1
2. Subset {1, 3}:
   Minor of A: det([1, 0; 0, 1]) = 1 - 0 = 1
   Minor of B: det([1, 0; 0, 2]) = 2 - 0 = 2
   Product = (1)(2) = 2
3. Subset {2, 3}:
   Minor of A: det([2, 0; 1, 1]) = 2 - 0 = 2
   Minor of B: det([2, 1; 0, 2]) = 4 - 0 = 4
   Product = (2)(4) = 8

The sum of these minor products is 1 + 2 + 8 = 11, perfectly matching det(AB).

Interactive Experiment

Explore the relationship between the rectangular factors and their minors in the Cauchy-Binet Formula module:

1. Observe the initial state with the matrices from the worked example, noting that the sum of the minor products equals 11.
2. Modify the entries of A (e.g., set A12 to -1). Notice how the minor determinants can become negative, causing partial cancellation in the sum.
3. Try creating a rank-one matrix for A by setting the second row to be a multiple of the first row (e.g., set A21=2, A22=4, A23=0). Observe that all 2×2 minors of A become zero, and thus det(AB) collapses to 0.
4. Watch the dynamic updates in the "Cauchy-Binet Expansion" panel to see exactly which minor products contribute to the final determinant.

Limitations and Edge Cases

While the Cauchy-Binet formula is a powerful generalization of the multiplicativity of the determinant, it has several limitations and degenerate cases depending on the dimensions and properties of the factor matrices.

• Square vs Rectangular: If A and B were square (e.g., 2×2), the formula simplifies to the familiar multiplicativity property: det(AB) = det(A)det(B).
• Oversized Factors: If A is 3×2 and B is 2×3, the product AB is 3×3. Because the rank of AB can be at most 2, its determinant is necessarily 0. In Cauchy-Binet terms, this occurs because there are no 3×3 minors to choose from the inner dimension (which is 2), so the sum is empty.
• Rank Deficiency: If either A or B does not have full column/row rank, some (or all) of its maximal minors will be zero, leading to a zero determinant for the product.

Common Pitfalls

Never write det(AB) = det(A)det(B) when A and B are rectangular matrices; determinants are only defined for square matrices. Moreover, individual 2×2 minor products can be positive or negative and can cancel completely even when both A and B have full rank 2.

References

The Cauchy–Binet formula is a foundational result in multi-linear algebra, generalizing the determinant of a product to rectangular matrices. This result was first independently discovered and proved by Augustin-Louis Cauchy and Jacques Philippe Marie Binet in the early 19th century.

• See: Introduction to the Cauchy-Binet formula (arXiv:1305.0644).