Multinomial functions of matrices

[Stephen Tashi]

TL;DR Summary

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 $f(A,B,C) = ABC + BAC + 2A^2 + 3C$)

 

[fresh_42]

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.

 

[WWGD]

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.

 

[Stephen Tashi]

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?