View source for Cauchy Binet formula

[[Category:Linear and multilinear algebra; matrix theory]]
{{MSC|15Axx|}}
{{TEX|done}}


A formula aimed at expressing the determinant of the product of two matrices $A\in\mathrm{M}_{m,n}(\mathbb{R})$ 
and $B\in\mathrm{M}_{n,m}(\mathbb{R})$, in terms of the sum of the products of all possible higher order [[Minor|minors]] 
of $A$ with corresponding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ 
denotes  any [[Multiindex|multi-index]] $(\beta_1,\ldots,\beta_m)$ with $1\leq  \beta_1<\ldots<\beta_m\leq n$ of 
length $m$, then
\[
\det(AB)=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\alpha},
\]
where $A_{\alpha\,\beta}=(a_{\alpha_i\beta_j})$ and $B_{\beta\,\alpha}=(a_{\beta_j\alpha_i})$.
In case $m>n$, no such $\beta$ exists and the right-hand side above is set to be $0$ by definition.

Note that if $n=m$ the formula reduces to 
\[
\det (AB)=\det A\,\det B.
\]
More generally, if $A\in\mathrm{M}_{m,n}(\mathbb{R})$, $B\in\mathrm{M}_{n,q}(\mathbb{R})$ 
and $p\leq\min\{m,q\}$, then any minor of order $p$ of the product matrix $AB$ can be expressed 
as follows by Cauchy Binet formula
\[
\det((AB)_{\alpha\,\gamma})=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\gamma},
\]
where $\alpha=(\alpha_1\ldots,\alpha_p)$ with $1\leq\alpha_1<\ldots<\alpha_p\leq m$, 
$\gamma=(\gamma_1,\ldots,\gamma_p)$ with $1\leq\gamma_1<\ldots<\gamma_p\leq q$, and
$\beta=(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$.

A number of interesting consequence of Cauchy Binet formula is listed below.
First of all, an inequality for the [[Rank|rank]] of the product matrix 
follows straightforwardly, i.e.,
\[
\mathrm{rank}(AB)\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
\]
Moreover, if $m=2$, $\mathbf{a}$, $\mathbf{b}\in\mathbb{R}^n$ are two vectors, 
by taking
$$A=\begin{pmatrix}
a_{1}&\dots&a_{n}\\
b_{1}&\dots&b_{n}\\
\end{pmatrix}
\quad\text{and}\quad
B=\begin{pmatrix}
a_{1}&b_{1}\\
\dots&\dots\\
a_{n}&b_{n}\\
\end{pmatrix}
$$
the Cauchy Binet formula yields
\[
\sum_{1\leq i<j\leq n}\begin{vmatrix}
a_{i}&a_{j}\\
b_{i}&b_{j}\\
\end{vmatrix}^2=
\begin{vmatrix}
\|\mathbf{a}\|^2&\langle\mathbf{a},\mathbf{b}\rangle\\
\langle\mathbf{a}, \mathbf{b}\rangle&\|\mathbf{b}\|^2\\
\end{vmatrix},
\]
in turn implying the [[Cauchy Schwartz inequality]]. Here, $\|\cdot\|$ and 
$\langle\cdot,\cdot\rangle$ are the Euclidean norm and scalar product, respectively.

Let us finally interpret geometrically the result. Take $B=A^T$, then 
$\det(A_{\alpha\beta})=\det(A^T_{\beta,\alpha})$, so that by Cauchy Binet formula
\[\label{p}
\det(A^T\,A)=\sum_\beta(\det(A_{\alpha\beta}))^2.
\]
This is a generalization of the Pythagorean formula, corresponding to $m=1$. Indeed, 
if $\mathcal{B}:\mathbb{R}^m\to\mathbb{R}^n$ is the linear map associated to $A^T$, 
and $Q\subset\mathbb{R}^m$ is the unitary cube, the $m$-th dimensional volume of the 
parallelepiped $\mathcal{A}(Q)\subset\mathbb{R}^n$ is given by $\sqrt{\det(A^T\,A)}$ 
due to the [[Polar decomposition|polar decomposition]] of $A$, recall that $m\leq n$.

Formula (1) above then expresses the square of the $m$-th dimensional volume of 
$\mathcal{A}(Q)$ as the sum of the squares of the volumes of the projections on 
all coordinates $m$ planes (cp. with [[Area formula|Area formula]]).


===References===
{|
|-
|valign="top"|{{Ref|EG}}||    L.C. Evans, R.F. Gariepy,  "Measure theory   and fine properties of    functions" Studies in Advanced  Mathematics.  CRC  Press, Boca Raton,    FL,  1992. {{MR|1158660}}  {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}|| 
F.R. Gantmacher, "The theory of matrices. Vol. 1", AMS Chelsea Publishing, Providence, RI, (1998).
{{MR|1657129}}
|-
|}