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The Cayley-Hamilton theorem for supermatrices. (English) Zbl 0834.15026

Summary: Starting from the expression for the superdeterminant of \((xI- M)\), where \(M\) is an arbitrary supermatrix, we propose a definition for the corresponding characteristic polynomial and we prove that each supermatrix satisfies its characteristic equation. Depending upon the factorization properties of the basic polynomials whose ratio defines the above-mentioned superdeterminant we are able to construct polynomials of lower degree which are also shown to be annihilated by the supermatrix. Some particular cases and examples are discussed.

MSC:

15A75 Exterior algebra, Grassmann algebras
15A24 Matrix equations and identities
15A90 Applications of matrix theory to physics (MSC2000)
81T60 Supersymmetric field theories in quantum mechanics
83E99 Unified, higher-dimensional and super field theories
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