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

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[[Category:Linear and multilinear algebra; matrix theory]]
 
[[Category:Linear and multilinear algebra; matrix theory]]
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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 $C=A\cdot B$, $A\in$\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  
 
$A$ with corresponding minors of the same order of $B$.
 
$A$ with corresponding minors of the same order of $B$.
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It follows straightforwardly an inequality for the [[Rank|rank]] of the product matrix, i.e.,
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\[
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\mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}.
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\]

Revision as of 14:12, 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 $C=A\cdot B$, $A\in$\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 $A$ with corresponding minors of the same order of $B$.


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=28817