Difference between revisions of "User:Matteo.focardi/sandbox"
From Encyclopedia of Mathematics
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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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| + | \mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}. | ||
| + | \] | ||
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
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28817