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The permanent function

University of British Columbia

Let X be a square matrix of order k over a field F. The permanent of X is given by [Formula omitted] where σ ranges over all the permutations of 1,2,...,k. The original object of this investigation was to characterize those linear maps which leave the permanent unaltered ; that is, per(X) = per(T(X)), all X. Let M[subscript m,n] denote the vector space of all matrices having m rows and n columns with entries taken from F. Fix an integer r, 2 ≤ r ≤ min(m,n). The r-th permanental compound of X ε M[subscript m,n] is defined in an analogous way to the r-th compound of X, and is denoted by P[subscript r](X) ε M[subscript (m over r) [comma] (n over r)]. Subject to mild restrictions on F, the following theorem can be proved. Let T be a linear map on M[subscript m,n] into itself, let S[subscript r] be a non-singular linear map on M[subscript (m over r) [comma] (n over r)] onto itself. Suppose that P[subscript r](T(X)) = S[subscript r](P[subscript r](X)), all X ε M[subscript m,n]. Then for max(m,n) > 2, we have T(X) = DPXQK when m ≠ n ; when m = n , we have either T(X) = DPXQK, allX, or T(X) = DPX'QK, all X. Here P,Q are permutation matrices and D,K are diagonal matrices, of appropriate orders. For the case r = m = n = 2 , there is a certain non-singular linear map B on M[subscript 2,2] onto itself such that BTB(X) = UXV, all X, or BTB(X) = UX'V, all X. Here U,V are non-singular. The original problem arises in the case r = m = n , with S[subscript r] =1, the unit of F.

Text

2012-01-19

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http://hdl.handle.net/2429/40186

Mathematics

University of British Columbia

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