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A formula aimed at expressing the determinant of a matrix $C\in M_{m,m}(\mathbb{R})$ that is the  
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A formula aimed at expressing the determinant of the product of two matrices $A\in\mathrm{M}_{m,n}(\mathbb{R})$  
product of $A\in\mathrm{M}_{m,n}(\mathbb{R})$ and $B\in\mathrm{M}_{n,m}(\mathbb{R})$, in terms of  
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and $B\in\mathrm{M}_{n,m}(\mathbb{R})$, in terms of the sum of the products of all possible higher order minors  
the sum of the products of all possible higher order minors of $A$ with corresponding minors of  
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of $A$ with corresponding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$  
the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ denotes any  
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denotes any [[Multiindex|multi-index]] $(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ of length $m$, then
[[Multiindex|multi-index]] $(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ of length $m$, then
 
 
\[
 
\[
\det C=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\alpha},
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\det(AB)=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\alpha},
 
\]
 
\]
 
where $A_{\alpha\,\beta}=(a_{\alpha_i\beta_j})$ and $B_{\beta\,\alpha}=(a_{\beta_j\alpha_i})$.
 
where $A_{\alpha\,\beta}=(a_{\alpha_i\beta_j})$ and $B_{\beta\,\alpha}=(a_{\beta_j\alpha_i})$.
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If $n=m$ the formula reduces to  
 
If $n=m$ the formula reduces to  
 
\[
 
\[
\det C=\det A\,\det B.
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\det (AB)=\det A\,\det B.
 
\]
 
\]
 
A number of interesting consequence of Cauchy-Binet formula is listed below.
 
A number of interesting consequence of Cauchy-Binet formula is listed below.
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\mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
 
\mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
 
\]
 
\]
Moreover, if $m=2$ and $\mathbf{a}$, $\mathbf{b}\in\mathbb{R}^n$ are two vectors then
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Moreover, if $m=2$, $\mathbf{a}$, $\mathbf{b}\in\mathbb{R}^n$ are two vectors,
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by taking
 
$$A=\begin{pmatrix}
 
$$A=\begin{pmatrix}
 
a_{1}&\dots&a_{n}\\
 
a_{1}&\dots&a_{n}\\
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\end{pmatrix}
 
\end{pmatrix}
 
$$
 
$$
then by Cauchy-Binet
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by Cauchy-Binet
 
\[
 
\[
 
\sum_{1\leq i<j\leq n}\begin{pmatrix}
 
\sum_{1\leq i<j\leq n}\begin{pmatrix}
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\begin{pmatrix}
 
\begin{pmatrix}
 
\|\mathbf{a}\|^2&\mathbf{a}\cdot \mathbf{b}\\
 
\|\mathbf{a}\|^2&\mathbf{a}\cdot \mathbf{b}\\
\mathbf{a}\cdot \mathbf{b}&\|\mathbf{b}\|^2\\
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\mathbf{a}\cdot \mathbf{b}&\|\mathbf{b}\|^2,\\
 
\end{pmatrix}
 
\end{pmatrix}
 
\]
 
\]
 
in turn implying Cauchy-Schwartz inequality.
 
in turn implying Cauchy-Schwartz inequality.
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If $A\in\mathrm{M}_{m,n}(\mathbb{R})$ and $B\in\mathrm{M}_{n,q}(\mathbb{R})$,
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and $N\leq\min\{m,q\}$ then any minor of order $N$ of the product matrix $C=AB$

Revision as of 14:35, 23 November 2012

2020 Mathematics Subject Classification: Primary: 15Axx [MSN][ZBL]


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 minors of $A$ with corresponding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ denotes any multi-index $(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ 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.

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

If $A\in\mathrm{M}_{m,n}(\mathbb{R})$ and $B\in\mathrm{M}_{n,q}(\mathbb{R})$, and $N\leq\min\{m,q\}$ then any minor of order $N$ of the product matrix $C=AB$

How to Cite This Entry:
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28827