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Difference between revisions of "User:Matteo.focardi/sandbox"

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A formula aimed at expressing the determinant of a square $m\times m$ matrix $C=A\cdot B$, $A\in$\def\M{\mathrm{M}}\M_{m,n}(\mathbb{R})$ and $A\in$\def\M_{n,m}(\mathbb{R})$, in terms the sum of products of all possible the higher order minors of  
+
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  
$A$ with corresponding minors of the same order of $B$.
+
of the products of all possible higher order minors of $A$ with corresponding minors of the  
 
+
same order of $B$.
 
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More precisely, if $\alpha=(1,\ldots,m)$ and $\beta$ denotes any [[Multiindex|multi-index]]
 +
$(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ of length $m$, then
 +
\[
 +
\det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha}.
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\]
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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.,
 
It follows straightforwardly an inequality for the [[Rank|rank]] of the product matrix, i.e.,

Revision as of 14:16, 23 November 2012

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


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 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 \[ \det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha}. \] 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 of the product matrix, i.e., \[ \mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}. \]

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