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{{MSC|28A}}
 
  
[[Category:Classical measure theory]]
 
  
{{TEX|done}}
 
  
  
A theorem on the relation between the concepts of  almost-everywhere convergence and uniform convergence of a sequence of  functions. Let $\mu$ be a [[Set function|$\sigma$-additive measure]]  defined on a [[Algebra of sets|$\sigma$-algebra]] <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351204.png" />, let <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e0351205.png" />,  $\mu(E)<+\infty$, and let a sequence of $\mu$-measurable  almost-everywhere finite functions $f_k(x)$, $x\in E$, $k=1,2,\ldots$,  converge almost-everywhere to a function $f(x)$. Then for any  $\varepsilon>0$ there exists a measurable set $E_\varepsilon\subset  E$ such that $\mu(E\setminus E_\varepsilon)<\varepsilon$, and the  sequence $f_k(x)$ converges to  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512016.png" /> uniformly on  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512017.png" />. For the case  where <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512018.png" /> is the Lebesgue  measure on the line this was proved by D.F. Egorov [[#References|[1]]].
 
  
Egorov's  theorem has various generalizations extending its potentialities. For  example, let <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512019.png" /> be a sequence of  measurable mappings of a locally compact space <img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512020.png" /> into a  metrizable space <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512021.png" /> for which the  limit
+
[[Category:Linear and multilinear algebra; matrix theory]]
 +
{{MSC|15Axx|}}
 +
{{TEX|done}}
  
<table class="eq"  style="width:100%;"> <tr><td valign="top"  style="width:94%;text-align:center;"><img align="absmiddle"  border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512022.png"  /></td> </tr></table>
 
  
exists  locally almost-everywhere on <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512023.png" /> with respect to a  [[Radon measure|Radon measure]] <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512024.png" />. Then <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512025.png" /> is measurable  with respect to <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512026.png" />, and for any  compact set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512027.png" /> and <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512028.png" /> there is a  compact set <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512029.png" /> such that <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512030.png" />, and the restriction of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512031.png" /> to <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512032.png" /> is continuous and  converges to <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512033.png" /> uniformly on  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512034.png" />. The conclusion  of Egorov's theorem may be false if <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512035.png" /> is not  metrizable.
+
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 [[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.
  
====References====
+
Note that if $n=m$ the formula reduces to
<table><TR><TD  valign="top">[1]</TD> <TD valign="top">  D.F. Egorov,    "Sur les suites de fonctions mesurables"  ''C.R. Acad. Sci. Paris'' ,  '''152'''  (1911) pp. 244–246  {{MR|}} {{ZBL|}}  </TD></TR><TR><TD valign="top">[2]</TD>  <TD valign="top">  A.N. Kolmogorov,   S.V. Fomin,   "Elements of  the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)  {{MR|1025126}} {{MR|0708717}{{MR|0630899}} {{MR|0435771}} {{MR|0377444}} {{MR|0234241}{{MR|0215962}} {{MR|0118796}} {{MR|1530727}} {{MR|0118795}}  {{MR|0085462}} {{MR|0070045}} {{ZBL|0932.46001}} {{ZBL|0672.46001}}  {{ZBL|0501.46001}} {{ZBL|0501.46002}} {{ZBL|0235.46001}}  {{ZBL|0103.08801}} </TD></TR><TR><TD  valign="top">[3]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Integration" , Addison-Wesley  (1975) pp.  Chapt.6;7;8  (Translated from French)  {{MR|0583191}} {{ZBL|1116.28002}}  {{ZBL|1106.46005}} {{ZBL|1106.46006}} {{ZBL|1182.28002}}  {{ZBL|1182.28001}} {{ZBL|1095.28002}} {{ZBL|1095.28001}}  {{ZBL|0156.06001}} </TD></TR></table>
+
\[
 +
\det (AB)=\det A\,\det B.
 +
\]
 +
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|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|polar decomposition]] of $A$, recall that $n\leq m$.
  
====Comments====
+
Formula (1) above then expresses the square of the $n$-th dimensional volume of
In  1970, G. Mokobodzki obtained a nice generalization of Egorov's theorem  (see [[#References|[a2]]], [[#References|[a3]]]): Let <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512036.png" />, <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512037.png" /> and <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512038.png" /> be as above. Let  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512039.png" /> be a set of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512040.png" />-measurable finite  functions that is compact in the topology of [[Pointwise  convergence|pointwise convergence]]. Then there is a sequence <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512041.png" /> of disjoint sets  belonging to <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512042.png" /> such that the support of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512043.png" /> is contained in  <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512044.png" /> and such that,  for every <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512045.png" />, the set <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512046.png" /> of restrictions  to <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512047.png" /> of the elements  of <img align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512048.png" /> is compact in the  topology of [[Uniform convergence|uniform convergence]].
+
$\mathcal{A}(Q)$ as the sum of the squares of the volumes of the projections on
 +
all coordinates $n$ planes (cp. with [[Area formula|Area formula]]).
  
Egorov's  theorem is related to the [[Luzin-C-property|Luzin <img  align="absmiddle" border="0"  src="https://www.encyclopediaofmath.org/legacyimages/e/e035/e035120/e03512049.png" />-property]].
 
  
====References====
+
===References===
<table><TR><TD  valign="top">[a1]</TD> <TD valign="top">  P.R. Halmos,    "Measure theory" , v. Nostrand  (1950)  {{MR|0033869}} {{ZBL|0040.16802}} </TD></TR><TR><TD  valign="top">[a2]</TD> <TD valign="top">  C. Dellacherie,   P.A. Meyer"Probabilities and potential" , '''C''' , North-Holland   (1988)  (Translated from French) {{MR|0939365}} {{ZBL|0716.60001}} </TD></TR><TR><TD valign="top">[a3]</TD>  <TD valign="top">  D. Revuz,   "Markov chains" , North-Holland  (1975) {{MR|0415773}} {{ZBL|0332.60045}}  </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|EG}}||  L.C. Evans, R.F. Gariepy, "Measure theory   and fine properties of  functions" Studies in Advanced  Mathematics.  CRC  Press, Boca RatonFL, 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=28507