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A formula aimed at expressing the determinant of a square $m\times m$ matrix $C=A\cdot B$, $A\in\mathrm{M}{m,n}(\mathbb{R})$ and $B\mathrm{M}_{n,m}(\mathbb{R})$, in terms of the sum  
+
A formula aimed at expressing the determinant of the product of two matrices $A\in\mathrm{M}_{m,n}(\mathbb{R})$  
of the products of all possible higher order minors of $A$ with corresponding minors of the  
+
and $B\in\mathrm{M}_{n,m}(\mathbb{R})$, in terms of the sum of the products of all possible higher order minors  
same order of $B$.
+
of $A$ with corresponding minors of the same order of $B$. More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$  
More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ denotes any [[Multiindex|multi-index]]
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denotes any [[Multiindex|multi-index]] $(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$ of length $m$, then
$(k_1,\ldots,k_m)$ 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},
 
\]
 
\]
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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.
 
In case $m>n$, no such $\beta$ exists and the right-hand side above is set to be $0$ by definition.
  
It follows straightforwardly an inequality for the [[Rank|rank]] of the product matrix, i.e.,
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Note that if $n=m$ the formula reduces to
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\[
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\det (AB)=\det A\,\det B.
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\]
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More generally, if $A\in\mathrm{M}_{m,n}(\mathbb{R})$, $B\in\mathrm{M}_{n,q}(\mathbb{R})$
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and $p\leq\min\{m,q\}$, then any minor of order $p$ of the product matrix $AB$ can be expressed
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as follows by Cauchy-Binet's formula
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\[
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\det((AB)_{\alpha\,\gamma})=\sum_\beta\det A_{\alpha\,\beta}\det B_{\beta\,\gamma},
 +
\]
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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
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$\beta=(\beta_1,\ldots,\beta_m)$ with $1\leq \beta_1<\ldots<\beta_m\leq n$.
 +
 
 +
A number of interesting consequence of Cauchy-Binet's formula is listed below.
 +
First of all, an inequality for the [[Rank|rank]] of the product matrix  
 +
follows straightforwardly, i.e.,
 
\[
 
\[
\mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
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\mathrm{rank}(AB)\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
 
\]
 
\]
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Moreover, if $m=2$, $\mathbf{a}$, $\mathbf{b}\in\mathbb{R}^n$ are two vectors,
 +
by taking
 +
$$A=\begin{pmatrix}
 +
a_{1}&\dots&a_{n}\\
 +
b_{1}&\dots&b_{n}\\
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\end{pmatrix}
 +
\quad\text{and}\quad
 +
B=\begin{pmatrix}
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a_{1}&b_{1}\\
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\dots&\dots\\
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a_{n}&b_{n}\\
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\end{pmatrix}
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$$
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Cauchy-Binet's formula yields
 +
\[
 +
\sum_{1\leq i<j\leq n}\begin{vmatrix}
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a_{i}&a_{j}\\
 +
b_{i}&b_{j}\\
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\end{vmatrix}^2=
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\begin{vmatrix}
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\|\mathbf{a}\|^2&\langle\mathbf{a},\mathbf{b}\rangle\\
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\langle\mathbf{a}, \mathbf{b}\rangle&\|\mathbf{b}\|^2\\
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\end{vmatrix},
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\]
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in turn implying Cauchy-Schwartz's inequality. Here, $\|\cdot\|$ and
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$\langle\cdot,\cdot\rangle$ are the Euclidean norm and scalar product, respectively.
 +
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Let us finally interpret geometrically the result. Take $B=A^T$, then
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$\det(A_{\alpha\beta})=\det(A^T_{\beta,\alpha})$, so that by Cauchy-Binet's formula
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\[\label{p}
 +
\det(A^T\,A)=\sum_\beta(\det(A_{\alpha\beta}))^2.
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\]
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This is a generalization of the Pythagorean formula, corresponding to $m=1$. Indeed,
 +
if $\mathcal{B}:\mathbb{R}^n\to\mathbb{R}^m$ is the linear map associated to $A^T$,
 +
and $Q\subset\mathbb{R}^n$ is the unitary cube, the $n$-th dimensional volume of the
 +
parallelepiped $\mathcal{A}(Q)\subset\mathbb{R}^m$ is given by $\sqrt{\det(A^T\,A)}$
 +
due to [[Polar decomposition|polar decomposition]] of $A$, recall that $n\leq m$.
 +
 +
Formula (1) above then expresses the square of the $n$-th dimensional volume of
 +
$\mathcal{A}(Q)$ as the sum of the squares of the volumes of the projections on
 +
all coordinates $n$ 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}}
 +
|-
 +
|}

Latest revision as of 16:10, 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 $(\beta_1,\ldots,\beta_m)$ 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's 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's formula is listed below. First of all, an inequality for the 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 $$A=\begin{pmatrix} a_{1}&\dots&a_{n}\\ b_{1}&\dots&b_{n}\\ \end{pmatrix} \quad\text{and}\quad B=\begin{pmatrix} a_{1}&b_{1}\\ \dots&\dots\\ a_{n}&b_{n}\\ \end{pmatrix} $$ Cauchy-Binet's 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 Cauchy-Schwartz's 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's 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 $\mathcal{B}:\mathbb{R}^n\to\mathbb{R}^m$ is the linear map associated to $A^T$, and $Q\subset\mathbb{R}^n$ is the unitary cube, the $n$-th dimensional volume of the parallelepiped $\mathcal{A}(Q)\subset\mathbb{R}^m$ is given by $\sqrt{\det(A^T\,A)}$ due to polar decomposition of $A$, recall that $n\leq m$.

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


References

[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe]

F.R. Gantmacher, "The theory of matrices. Vol. 1", AMS Chelsea Publishing, Providence, RI, (1998). MR1657129

How to Cite This Entry:
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28820