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I am trying to find definition of determinant where the matrix entries may not be commutative, for instance quaternions or Clifford algebras. Does the minor expansion of determinant over a field still work over a non-commutative algebra?
I am trying to find definition of determinant where the matrix entries may not be commutative, for instance quaternions or Clifford algebras. Does the minor expansion of determinant over a field still work over a non-commutative algebra?
My gut says that a useful definition of determinant could be defined as follows:
Your algebra A is a finite dimensional algebra over the reals -- I think that implies that it can be represented as an algebra of real matrices. So you can view your matrix over A as a matrix over R, and take the (real-valued!) determinant of that.