Difference between revisions of "User:Matteo.focardi/sandbox"
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A formula aimed at expressing the determinant of a matrix $C\in M_{m,m}(\mathbb{R})$ that is the | A formula aimed at expressing the determinant of a matrix $C\in M_{m,m}(\mathbb{R})$ that is the | ||
| − | product of $A\in\mathrm{M}_{m,n}(\mathbb{R})$ and $B\mathrm{M}_{n,m}(\mathbb{R})$, in terms of | + | product of $A\in\mathrm{M}_{m,n}(\mathbb{R})$ 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 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 | 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 | [[Multiindex|multi-index]] $(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ of length $m$, then | ||
\[ | \[ | ||
| − | \det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha} | + | \det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha}, |
\] | \] | ||
| + | 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. | In case $m>n$, no such $\beta$ exists and the right-hand side above is set to be $0$ by definition. | ||
Moreover, if $n=m$ the formula reduce to | Moreover, if $n=m$ the formula reduce to | ||
Revision as of 14:21, 23 November 2012
2020 Mathematics Subject Classification: Primary: 15Axx [MSN][ZBL]
A formula aimed at expressing the determinant of a matrix $C\in M_{m,m}(\mathbb{R})$ that is the product of $A\in\mathrm{M}_{m,n}(\mathbb{R})$ 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 $(k_1,\ldots,k_m)$ with $1\leq k_1<\ldots<k_m\leq n$ of length $m$, then \[ \det C=\sum_\beta\det A_{\alpha\beta}\det B_{\beta\alpha}, \] 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. Moreover, if $n=m$ the formula reduce to \[ \det C=\det A\,\det B. \] A numberber of interesting consequence of Cauchy-Binet formula is listed below. First of all, we get straightforwardly an inequality for the rank of the product matrix $C$, 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=28822