Cauchy-Binet formula

In linear algebra, the Cauchy-Binet formula generalizes the multiplicativity of the determinant (the fact that the determinant of a product of two square matrices is equal to the product of the two determinants) to non-square matrices.

Suppose "A" is an "m"×"n" matrix and "B" is an "n"×"m" matrix. If "S" is a subset of { 1, ..., "n" } with "m" elements, we write "A"_"S" for the "m"×"m" matrix whose columns are those columns of "A" that have indices from "S". Similarly, we write "B"_"S" for the "m"×"m" matrix whose "rows" are those rows of "B" that have indices from "S". The Cauchy-Binet formula then states

:$det(AB)\; =\; sum\_S\; det(A\_S)det(B\_S),$

where the sum extends over all possible subsets "S" of { 1, ..., "n" } with "m" elements (there are C("n","m") of them).

If "m" = "n", i.e. if "A" and "B" are square matrices of the same format, then there is only a single admissible set "S", and the Cauchy-Binet formula reduces to the ordinary multiplicativity of the determinant. If "m" = 1 then there are "n" admissible sets "S" and the formula reduces to that for the dot product. If "m" > "n", then there is no admissible set "S" and the determinant det("AB") is zero (see empty sum).

The formula is valid for matrices with entries from any commutative ring. For the proof one writes the columns of "AB" as linear combinations of the columns of "A" with coefficients from "B", uses the multilinearity of the determinant, and collects the terms that belong to a single det("A"_"S") together by exploiting the anti-symmetry of the determinant. The coefficient of det("A"_"S") is seen to be det("B"_"S") using the Leibniz formula for the determinant. This proof does not use the multiplicativity of the determinant; rather, the proof establishes it.

If "A" is a real "m"×"n" matrix, then det("A" "A"^T) is equal to the square of the "m"-dimensional volume of the parallelepiped spanned in R^"n" 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 C("n","m")). The case "m" = 1 of this statement talks about the length of a line segment: it is nothing but the Pythagorean theorem.

The Cauchy-Binet formula can be extended in a straightforward way to a general formula for the minors of the product of two matrices. That formula is given in the article on minors.

Example: If and then the Cauchy-Binet formula gives the determinant:

:
cdotleft|egin{matrix}1&1\3&1end{matrix} ight
+left|egin{matrix}1&2\1&-1end{matrix} ight
cdotleft|egin{matrix}3&1\0&2end{matrix} ight
+left|egin{matrix}1&2\3&-1end{matrix} ight
cdotleft|egin{matrix}1&1\0&2end{matrix} ight
=-28.