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- W2013559215 abstract "Let [Mn(C)] denote the set of linear maps from the n X n complex matrices into themselves and let Qn denote the set of complex doubly stochastic matrices, i.e. complex matrices whose row and column sums are 1. If FE[Mn(C)] is such that F(Qn) CQn and F*(U.) CO., then there exist Ai, Bi, A, and B EE Q such that F(X) = E AiXBi + AXtJn + J,XtB (1 + m)JnXJn for all n X n complex matrices X, where Jn is the n X n matrix whose elements aie each 1/n and where the superscript t denotes transpose. m denotes the number of the Ai (or Bi). Introduction. It has been of considerable interest to study linear maps from the n X n matrices to themselves that leave certain quantities invariant [1][12]. Often these maps are necessarily of the form F(X) =AXB or AXtB with certain restrictions imposed on the n Xn matrices A and B, where the superscript t denotes transpose. For example, Marcus and Moyls [8] show that such maps which preserve spectral values are of these forms with A unimodular and B =A-'. They show in [8], [9] that such maps which preserve certain given ranks are of these forms with A and B nonsingular. Marcus and May [7] show that such maps which preserve the permanent function are of these forms with A =P,Di and B =P2D2 where the Pi are permutation matrices and the Di are diagonal matrices such that per D1D2= 1. Marcus, Minc, and Moyls [10] show that one may assume that Di = D2=I if in addition the linear map leaves the doubly stochastic matrices invariant. This paper is concerned with linear transformations which map the set of n X n generalized doubly stochastic matrices, i.e. n X n complex matrices whose row and column sums are one, into itself. It is shown that the set of such maps F which includes both F and F* is precisely the set of linear combinations of transformations of the types AXB and CXtD, where the sum of the coefficients in any such Received by the editors May 12, 1969. AMS 1968 subject classifications. Primary 1565, 1585; Secondary 1530." @default.
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- W2013559215 date "1971-02-01" @default.
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- W2013559215 title "Linear transformations under which the doubly stochastic matrices are invariant" @default.
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