Binet-Cauchy's theorem

The Cauchy-Binet formula is a sentence from the mathematical branch Linear Algebra . The theorem, named after Jacques Philippe Marie Binet and Augustin-Louis Cauchy , consists of a formula for calculating the determinant of a square matrix . In order to use it, a product representation must be known. Binet-Cauchy's theorem generalizes the determinant product theorem , which results as a special case if and are quadratic. ${\displaystyle C}$ ${\displaystyle C=A\cdot B}$ ${\displaystyle A}$ ${\displaystyle B}$

sentence

If there is an -Matrix and an -Matrix, then the determinant of is calculated by adding up the products of a -dimensional minor of and : ${\displaystyle A}$ ${\displaystyle n\times m}$ ${\displaystyle B}$ ${\displaystyle m\times n}$ ${\displaystyle A\cdot B}$ ${\displaystyle n}$ ${\displaystyle A}$ ${\displaystyle B}$

${\displaystyle \det(A\cdot B)=\sum _{S\subseteq \{1,2,\ldots ,m\} \atop |S|=n}\det(A_{S})\det(B_{S})=\sum _{S\subseteq \{1,2,\ldots ,m\} \atop |S|=n}\det(A_{S}\cdot B_{S})}$

The sub-matrices and result from the matrices and if only the columns from or rows from are used whose numbers appear in . The original order of the columns or rows must be retained. If , then there are no such sub-matrices and it applies . ${\displaystyle A_{S}}$ ${\displaystyle B_{S}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle S}$ ${\displaystyle n>m}$ ${\displaystyle \det(A\cdot B)=0}$

If then there is exactly one subset and it applies . ${\displaystyle A,B\in \mathbb {R} ^{n\times n}}$ ${\displaystyle S=\{1,2,\ldots ,n\}}$ ${\displaystyle \det(A\cdot B)=\det(A)\cdot \det(B)}$

example

In this example the determinant of the matrix is calculated using Binet-Cauchy's theorem. The following product representation is given for this matrix: ${\displaystyle C}$

${\displaystyle C={\begin{pmatrix}58&64\\139&154\end{pmatrix}}={\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}}\cdot {\begin{pmatrix}7&8\\9&10\\11&12\end{pmatrix}}=A\cdot B}$.

According to Binet-Cauchy's theorem:

${\displaystyle \det(C)=\sum _{S\subseteq \{1,2,3\} \atop |S|=2}\det(A_{S})\det(B_{S})}$

${\displaystyle \qquad =\det(A_{\{1,2\}})\cdot \det(B_{\{1,2\}})+\det(A_{\{2,3\}})\cdot \det(B_{\{2,3\}})+\det(A_{\{1,3\}})\cdot \det(B_{\{1,3\}})}$

${\displaystyle \qquad =\det {\begin{pmatrix}1&2\\4&5\end{pmatrix}}\cdot \det {\begin{pmatrix}7&8\\9&10\end{pmatrix}}+\det {\begin{pmatrix}2&3\\5&6\end{pmatrix}}\cdot \det {\begin{pmatrix}9&10\\11&12\end{pmatrix}}+\det {\begin{pmatrix}1&3\\4&6\end{pmatrix}}\cdot \det {\begin{pmatrix}7&8\\11&12\end{pmatrix}}}$

${\displaystyle \qquad =(-3)\cdot (-2)+(-3)\cdot (-2)+(-6)\cdot (-4)}$

${\displaystyle \qquad =36}$.

literature

• Felix R. Gantmacher : Matrix theory. Springer-Verlag, 1986, ISBN 3-540-16582-7 , pp. 28-29
• Igor R. Shafarevich, Alexey O. Remizov: Linear Algebra and Geometry , Springer, 2012, ISBN 978-3-642-30993-9 , §2.9 (p. 68) & §10.5 (p. 377)