A Multinomial functions of matrices

  • A
  • Thread starter Thread starter Stephen Tashi
  • Start date Start date
  • Tags Tags
    Functions Matrices
Stephen Tashi
Science Advisor
Homework Helper
Education Advisor
Messages
7,864
Reaction score
1,602
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##)
 
Physics news on Phys.org
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.
 
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.
 
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}##?

-----
[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?
 
Last edited:
##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
Back
Top