Multinomial functions of matrices

  • #1
Stephen Tashi
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What branch of mathematics studies multinomial functions of matrices?
What branch of mathematics studies multinomial functions of matrices? ( i.e matrix valued functions of square matrices such as ##f(A,B,C) = ABC + BAC + 2A^2 + 3C##)
 

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  • #2
fresh_42
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Linear algebra if we are talking about scalar fields, abstract algebra if it is a ring. Functional analysis if the matrices are possibly infinite-dimensional linear operators, Lie theory if the matrices are part of a Lir group, topology if continuity is the main property, algebraic geometry if the zeroes of ##f## are the subject of interest.
 
  • #3
WWGD
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If I understand your question correctly, the Borel Calculus and other types of functional calculus provide a rigorous framework for applying standard "Calculus-like" functions to linear operators so that you can define , e.g., expressions like ##e^{A} ##; ##A## a matrix. I only remember minor details.
 
  • #4
Stephen Tashi
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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?
 
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