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Open Access Publications from the University of California

Extending Differential Calculus for Noncommuting Variables

  • Author(s): Schwartz, Charles
  • et al.

Recent innovations in the  differential calculus for functions of non-commuting variables, begun for a quaternionic variable,are readily extended to the case of a general Clifford algebra and then extended more to the case of a variable that is a general square matrix over the complex numbers. The expansion of F(x+delta) is given to first order in delta for general matrix variables x and delta that do not commute with each other. Further extension leads to the broader study of   commutators with functions, [y, f(x)] , expressed in terms of [y, x].

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