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 | + | 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 $B$. | |
| − | + | 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 | ||
| + | \[ | ||
| + | \det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha}. | ||
| + | \] | ||
| + | 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., \[ \mathrm{rank}C\leq\min\{\mathrm{rank}A,\mathrm{rank}B\}. \]
Matteo.focardi/sandbox. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Matteo.focardi/sandbox&oldid=28820