Cauchy–Binet formula

The set of Binet - Cauchy is a set from the mathematical branch Linear Algebra. Named after Jacques Philippe Marie Binet and Augustin- Louis Cauchy set consists of a formula to calculate the determinant of a square matrix. To apply it, a product presentation must be known. The set of Binet - Cauchy generalizes the determinants of product set that arises as a special case when and are square.

Set

Are a matrix and a matrix, then the determinant of calculated by summing products of one -dimensional minor of and:

The sub-matrices and arise from the matrices and only if the columns or rows of used occur in their numbers. However, the original order of the columns or rows must be preserved. Is, then there is no such sub-matrices, and it is.

Are both matrices, then there is exactly one subset, and it is.

Example

In this example, the determinant of the matrix is ​​calculated by means of the set of Cauchy - Binet. For this matrix the following product representation is given:

By the theorem of Cauchy - Binet applies: