caa a22 4500 46582546X CHVBK 20180323112147.0 cr unu---uuuuu 170327e19901201xx s 000 0 eng 10.1007/BF02112290 doi (NATIONALLICENCE)springer-10.1007/BF02112290 The Binet-Pexider functional equation for rectangular matrices [Elektronische Daten] [K. Heuvers, D. Moak] Summary: IfK is a field of characteristic 0 then the following is shown. Iff, g, h: M n (K) →K are non-constant solutions of the Binet—Pexider functional equation $$f(AB) = \frac{1}{{n!}}\sum\limits_{\left| s \right| = n} {\left( {\begin{array}{*{20}c} n \\ {s_1 \cdots s_{n{\mathbf{ }} + {\mathbf{ }}r} } \\ \end{array} } \right)g(A^s )h(B_s )}$$ for rectangular matricesA ∈ M n ×(n + r) (K) andB ∈ M (n + r) ×n (K), and a fixed non-negative integerr thenf(X) = ab d(X), g(X) = a d(X), andh(X) = b d(X) wherea andb are arbitrary constants fromK andd: M n (K) →K is a non-constant solution of the Binet—Cauchy functional equation $$d(AB) = \frac{1}{{n!}}\sum\limits_{\left| s \right| = n} {\left( {\begin{array}{*{20}c} n \\ {s_1 \cdots s_m } \\ \end{array} } \right)d(A^s )d(B_s )}$$ forA ∈ M n ×m (K) andB ∈ M M ×n (K) wheren ⩽ m ⩽ n + r. The general non-constant solution to (2) has been shown by Heuvers, Cummings, and K. P. S. bhaskara Rao, to bed(X) = φ(perX), whereφ is an isomorphism ofK, provided thatd(E) ≠ 0 for then × n matrixE with all entries 1/n. Heuvers and Moak have shown that the general non-constant solution to (2) is given byd(x) = μ(detX), whereμ is a non-constant multiplicative function onK ifm = n andd(E) = 0. Ifn ⩽ m ⩽n + 1 andd(E) = 0 they have shown that it is given byd(X) = φ(detX), where φ is an isomorphism ofK. Birkhäuser Verlag, 1990 Primary 15A15, 39B60 nationallicence Heuvers K. Michigan Technological University, 49931, Houghton, MI, USA aut Moak D. Michigan Technological University, 49931, Houghton, MI, USA aut aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 136-146 0001-9054 40:1<136 1990 40 10 https://doi.org/10.1007/BF02112290 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF02112290 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Heuvers K. Michigan Technological University, 49931, Houghton, MI, USA aut NATIONALLICENCE 700 1- Moak D. Michigan Technological University, 49931, Houghton, MI, USA aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 40/1(1990-12-01), 136-146 0001-9054 40:1<136 1990 40 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer