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Capelli Identity
In mathematics, Capelli's identity, named after , is an analogue of the formula det(''AB'') = det(''A'') det(''B''), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra \mathfrak_n. It can be used to relate an invariant ''ƒ'' to the invariant Ω''ƒ'', where Ω is Cayley's Ω process. Statement Suppose that ''x''''ij'' for ''i'',''j'' = 1,...,''n'' are commuting variables. Write ''E''ij for the polarization operator :E_ = \sum_^n x_\frac. The Capelli identity states that the following differential operators, expressed as determinants, are equal: : \begin E_+n-1 & \cdots &E_& E_ \\ \vdots& \ddots & \vdots&\vdots\\ E_ & \cdots & E_+1&E_ \\ E_ & \cdots & E_& E_ +0\end = \begin x_ & \cdots & x_ \\ \vdots& \ddots & \vdots\\ x_ & \cdots & x_ \end \begin \frac & \cdots &\frac \\ \vdots& \ddots & \vdots\\ \frac & \cdots &\frac \end. Both sides are differential operators. The determinant on the left has non-co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
