Formulate of Binet-Cauchy

In linear algebra, the formula of Binet-Cauchy generalizes the property of multiplicativity of the determinant of a product to the rectangular cases of two matrices. One can write it for matrices whose coefficients are in a body, or more generally in a commutative Anneau.

Statement

To be able to carry out the product of the matrices has and B , one supposes that they are respective sizes m by N and N by m . The formula of Binet-Cauchy states then

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

in this expression S describes the various subsets with m elements of the unit {1,…, N }. For each S , the matrix has _S is the matrix of size m obtained by not retaining that the columns of has whose index belongs to S . Of the same B _S is the matrix of size m obtained by retaining only the lines of B whose index belongs to S .

If m=n , the matrices has and B is square, there is only one term in the formula of Binet-Cauchy, which gives again well the property of multiplicativity of the determinants.

If m > N , it does not have there overall S suitable and the determinant of AB is null, according to usual conventions on the empty sums.

If m < N , the formula requires to carry out the sum of $\backslash \; begin\; \{pmatrix\}\; N\; \backslash \; \backslash \; m\; \backslash \; end\; \{pmatrix\}$ terms.

One can write a more general form of the formula of Binet-Cauchy for the minor of a matrix.

Demonstration

One writes has in the form of a list of columns: A₁. , A_n , and B by detailing all the coefficients. The product AB is thus, in columns, of the form

$AB=\; \backslash \; left\; (\backslash \; sum\_\; \{j=1\}\; ^n\; b\_\; \{j1\}\; A\_1,\; \backslash \; dowries,\; \backslash \; sum\_\; \{j=1\}\; ^n\; b\_\; \{jm\}\; A\_n\; \backslash \; right)$

It is necessary to exploit the multilinearity determinant, and to gather the terms corresponding to same the det ( has _S ) by using the alternate character. The coefficient in front of det ( has _S ) is identified with det ( B _S ) by recognizing the formula of Leibniz.

This proof can be used to establish the property of product of the determinants (a more geometrical version was established in the article determinant).

Euclidean interpretation

If has is a real matrix of size m by N , then the determinant of the matrix has ^tA is equal to the square of volume m - dimensional of the Parallélotope generated in $\backslash \; mathbb\; \{R\}\; ^n$ by the m columns of has.

The formula of Binet-Cauchy shows that this quantity is equal to the sum of the squares of volumes of the orthogonal projections on the various subspaces of coordinates of dimension m (which are with the number of $\backslash \; begin\; \{pmatrix\}\; N\; \backslash \; \backslash \; m\; \backslash \; end\; \{pmatrix\}$).

In the case m=1 , these orthogonal projections are segments, and one finds a form of the Théorème of Pythagore.

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