Noteworthy Uses:: computing the [[volume of a parallelopiped]] Proved By:: Lots of [[determinant]] properties ----- - Let $\ff$ be a [[field]] (or any [[commutative ring]]) - Let $A\in \ff^{k \times n}$ and $B \in \ff^{n \times k}$ be [[matrix|matrices]]. > [!proposition] Proposition. ([[Cauchy-Binet identity]]) > The [[determinant]] of $AB$, $\det AB$, is given by the following sum over [[set of all ascending k-tuples from 1 to n]]: $\det AB = \sum_{I \in \asc_{k,n}} \det (A^{I})\det(B_I),$ where for $I=\{ i_{1},\dots,i_{k} \}$, $A^{I}$ is the $k$-by-$k$ [[submatrix]] of $B$ containing the *columns* $i_{1},\dots,i_{k}$, and $B_I$ is the $k$-by-$k$ [[submatrix]] of $B$ containing the *rows* $i_{1},\dots,i_{k}$. \ As a **corollary** we have the following generalization of the [[Pythagorean Theorem]]: For $k \leq n$ and $X \in \ff ^{n \times k}$, $V^{2}(X)=\det(X ^{\top}X)=\sum_{I \in\asc_{k,n}}^{n} (\det X_{I}) ^{2}.$ \ Another **corollary** is the [[multiplicativity of the determinant]]: $\det$ is a [[group homomorphism]] from the [[general linear group]] $GL_{n}(\mathbb{R}) \mapsto \mathbb{R}^{\times}$. > [!proof]- Proof. ([[Cauchy-Binet identity]]) > ![[CleanShot 2023-01-08 at [email protected]]] > ![[CleanShot 2023-01-08 at 22.45.29.jpg]] > [!basicexample]- > ![[CleanShot 2023-01-08 at 20.19.12.jpg]] > ###### Proof when $k=2$. > ![[CleanShot 2023-01-08 at 21.53.31.jpg]] > [!intuition]- > In a low-dimension case, compare with the expansion of the [[cross product]]. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```