In linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that the determinant of a product of square matrices is equal to the product of their determinants. The formula is valid for matrices with entries from any commutative ring.
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Statement
Let A be an m×n matrix and B an n×m matrix. Write [n] for the set { 1, ..., n }, and
Example: taking m = 2 and n = 3, and matrices
Indeed
Special cases
If n < m then
For m = 0, A and B are empty matrices (but of different shapes if n > 0), as is their product AB; the summation involves a single term S = Ø, and the formula states 1 = 1, with both sides given by the determinant of the 0×0 matrix. For m = 1, the summation ranges over the collection
In the case n = 3
Let
In the case m > 3, the right handed side always equals to 0.
A simple proof
The following simple proof presented in relies on two facts that can be proven in several different ways:
- For any
1 ≤ k ≤ n the coefficient ofz n − k det ( z I n + X ) is the sum of thek × k principal minors ofX . - If
m ≤ n andA is anm × n matrix andA ann × m matrix, thendet ( z I n + B A ) = z n − m det ( z I m + A B ) .
Now, if we compare the coefficient of
Proof
There are various kinds of proofs that can be given for the Cauchy−Binet formula. The proof below is based on formal manipulations only, and avoids using any particular interpretation of determinants, which may be taken to be defined by the Leibniz formula. Only their multilinearity with respect to rows and columns, and their alternating property (vanishing in the presence of equal rows or columns) are used; in particular the multiplicative property of determinants for square matrices is not used, but is rather established (the case n = m). The proof is valid for arbitrary commutative coefficient rings.
The formula can be proved in two steps:
- use the fact that both sides are multilinear (more precisely 2m-linear) in the rows of A and the columns of B, to reduce to the case that each row of A and each column of B has only one non-zero entry, which is 1.
- handle that case using the functions [m]→[n] that map respectively the row numbers of A to the column number of their nonzero entry, and the column numbers of B to the row number of their nonzero entry.
For step 1, observe that for each row of A or column of B, and for each m-combination S, the values of det(AB) and det(A[m],S)det(BS,[m]) indeed depend linearly on the row or column. For the latter this is immediate from the multilinear property of the determinant; for the former one must in addition check that taking a linear combination for the row of A or column of B while leaving the rest unchanged only affects the corresponding row or column of the product AB, and by the same linear combination. Thus one can work out both sides of the Cauchy−Binet formula by linearity for every row of A and then also every column of B, writing each of the rows and columns as a linear combination of standard basis vectors. The resulting multiple summations are huge, but they have the same form for both sides: corresponding terms involve the same scalar factor (each is a product of entries of A and of B), and these terms only differ by involving two different expressions in terms of constant matrices of the kind described above, which expressions should be equal according to the Cauchy−Binet formula. This achieves the reduction of the first step.
Concretely, the multiple summations can be grouped into two summations, one over all functions f:[m] → [n] that for each row index of A gives a corresponding column index, and one over all functions g:[m] → [n] that for each column index of B gives a corresponding row index. The matrices associated to f and g are
where "
where p(f,g) denotes the scalar factor
For this step 2, if f fails to be injective then Lf and LfRg both have two identical rows, and if g fails to be injective then Rg and LfRg both have two identical columns; in either case both sides of the identity are zero. Supposing now that both f and g are injective maps [m] → [n], the factor
Let h be the unique increasing bijection [m] → S, and π,σ the permutations of [m] such that
Using multi-linearity with respect to both the rows of A and the columns of B in the proof is not necessary; one could use just one of them, say the former, and use that a matrix product LfB either consists of a permutation of the rows of Bf([m]),[m] (if f is injective), or has at least two equal rows.
Relation to the generalized Kronecker delta
As we have seen, the Cauchy–Binet formula is equivalent to the following:
where
In terms of generalized Kronecker delta, we can derive the formula equivalent to the Cauchy–Binet formula:
Geometric interpretations
If A is a real m×n matrix, then det(A AT) is equal to the square of the m-dimensional volume of the parallelotope spanned in Rn by the m rows of A. Binet's formula states that this is equal to the sum of the squares of the volumes that arise if the parallelepiped is orthogonally projected onto the m-dimensional coordinate planes (of which there are
In the case m = 1 the parallelotope is reduced to a single vector and its volume its length. The above statement then states that the square of the length of a vector is the sum of the squares of its coordinates; this is indeed the case by the definition of that length, which is based on the Pythagorean theorem.
Generalization
The Cauchy–Binet formula can be extended in a straightforward way to a general formula for the minors of the product of two matrices. Context for the formula is given in the article on minors, but the idea is that both the formula for ordinary matrix multiplication and the Cauchy-Binet formula for the determinant of the product of two matrices are special cases of the following general statement about the minors of a product of two matrices. Suppose that A is an m × n matrix, B is an n × p matrix, I is a subset of {1,...,m} with k elements and J is a subset of {1,...,p} with k elements. Then
where the sum extends over all subsets K of {1,...,n} with k elements. This formula, further explained in the article on minors is a straightforward extension of the Cauchy–Binet formula.