I see that a multinomial function of square matrices amounts to "simultaneous" multinomial functions of the entries of the matrices. For example, if we have 2x2 matrices ##A,B## and the 2x2 matrix ##F## is a multinomial in ##A,B## then ##F_{1,2}## is a multinomial function of ##A_{1,1}, A_{1,2}, A_{2,1},A_{2,2}, B_{1,1}, B_{1,2}, B_{2,1}, B_ {2,2}##.
But is the converse true? i.e. If we are given 4 arbitrary multinomial functions ##F_{i,j} ## of those variables, can we find a matrix multinomial function in ##A,B## that gives identical ##F_{i,j}##?
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[An edit , added later:]
When applied to variables that are matrices, the terminology "multinomial" may require some clarification. For example, if the variables are ##X,Y## and ##C## is a constant matrix, then ## CXY##, ##XCY## and ##XYC## may be different functions. Do we wish to allow all three examples to be multinomial functions of matrices? - or do we wish to restrict the definition of a "multinomial" function of matrices so that constant (matrix) factors can only appear as the leading factor, or perhaps insist that constant factors must be multiples of the identity matrix?