The Multiplication Table is a Hermitian Matrix

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Discussion Overview

The discussion revolves around the properties of a 12 by 12 multiplication table represented as a matrix, specifically examining its characteristics as a Hermitian matrix and whether it has any physical significance or represents an observable in a physical system.

Discussion Character

  • Exploratory
  • Debate/contested
  • Conceptual clarification

Main Points Raised

  • One participant notes that the diagonal entries of the multiplication table are real numbers (squares of integers), while suggesting that the off-diagonal elements are real and complex conjugates of each other.
  • Another participant questions the presence of complex numbers in the multiplication table, asserting that the entries are real numbers.
  • There is a discussion about the definition of the multiplication table and the context in which it is used, particularly in early education for whole numbers.
  • Some participants express skepticism about the relevance of the multiplication table as a Hermitian matrix in relation to physical observables, with one suggesting that it does not correspond to any useful physical system.
  • One participant humorously suggests that the observable associated with the matrix could be related to the number of lives of Schrödinger's cat.
  • Another participant concludes that the discussion validates the initial guess that the matrix has no significant meaning in the context of Hermitian operators.

Areas of Agreement / Disagreement

Participants express differing views on the significance of the multiplication table as a Hermitian matrix, with some arguing it has no relevance to physical observables while others explore its properties without reaching a consensus.

Contextual Notes

The discussion highlights uncertainties regarding the definitions and implications of the multiplication table, particularly in relation to complex numbers and physical systems. There are unresolved questions about the applicability of the matrix in a broader mathematical or physical context.

jaurandt
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I was drawing out the multiplication table in "matrix" form (a 12 by 12 matrix) for a friend trying to pass the GED (yes, sad, I know) and noticed for the first time that the entries on the diagonal are real, i.e. the squares (1, 4, 9, 16, ...), and the off diagonal elements are real and complex conjugates of each other.

Since Hermitian operators or matrices are usually associated with some observable, I wondered, what might the 12 by 12 matrix of multiplication products represent in this sense? I'm guessing the answer is "nothing", but I just wanted to see...
 
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What's a "multiplication table/matrix"? Maybe, it's stupid to ask, but you haven't given a clear definition of the matrix you are considering.
 
##M_{ij}=ij##, where, in this case, ##i,j=1,\cdots,12##.
 
Haborix said:
##M_{ij}=ij##, where, in this case, ##i,j=1,\cdots,12##.
What did you multiply?
 
fresh_42 said:
What did you multiply?
I hope ##i## and ##j##, ha ha. I was trying to give a simple definition of the multiplication table that jaurandt is referring to.
 
Haborix said:
I hope ##i## and ##j##, ha ha. I was trying to give a simple definition of the multiplication table that jaurandt is referring to.
Sorry, confused you with the OP.

Multiplication table, o.k., but of what? The word multiplication table only allows to assume that there is a binary operation. But on which group, space, algebra, field or whatever?
 
fresh_42 said:
Sorry, confused you with the OP.

Multiplication table, o.k., but of what? The word multiplication table only allows to assume that there is a binary operation. But on which group, space, algebra, field or whatever?
What the OP is referring to is something used in early education in the U.S.. This is really something one would learn between the ages of 6 and 9 (I don't remember exactly when) to multiply whole numbers. I interpret the OP to be asking if there is any physical system or example for which, by coincidence, this multiplication table makes an appearance.
 
To specify, yes a 12 by 12 matrix where elementij = i*j where elementsii = i*i (1, 4, 9, 16, ... , 144)

It is a Hermitian matrix and I was in fact wondering if it could apply to some observable of a system.
 
jaurandt said:
I was drawing out the multiplication table in "matrix" form (a 12 by 12 matrix) for a friend trying to pass the GED (yes, sad, I know) and noticed for the first time that the entries on the diagonal are real, i.e. the squares (1, 4, 9, 16, ...),
Well, yes, of course. Each entry on the diagonal consists of one of the numbers 1 through 12 being multiplied by itself.
jaurandt said:
and the off diagonal elements are real and complex conjugates of each other.
?
What are you talking about? There are no complex numbers in the multiplication table, unless you want to write a number as, say, 3 + 0i. I don't know why you would want to do that.
 
  • #10
jaurandt said:
It is a Hermitian matrix and I was in fact wondering if it could apply to some observable of a system.
Only if the vectors to which you are applying this operator correspond to the states of some physical system. In this case they don't seem to, or at least I can't think of any physical system that might be usefully characterized by these vectors.
 
  • #11
Mark44 said:
Well, yes, of course. Each entry on the diagonal consists of one of the numbers 1 through 12 being multiplied by itself.
?
What are you talking about? There are no complex numbers in the multiplication table, unless you want to write a number as, say, 3 + 0i. I don't know why you would want to do that.

As far as I know, a real number is it's own complex conjugate, just like 0.
 
  • #12
jaurandt said:
As far as I know, a real number is it's own complex conjugate, just like 0.
Sure. My example was 3 + 0i, which is certainly equal to 3 - 0i, but so what?
 
  • #13
jaurandt said:
As far as I know, a real number is it's own complex conjugate, just like 0.
That is true, but as long as you're confining yourself to operations that are closed on the real numbers there's not a lot of insight to be gained by treating your reals as complex... so people tend to think of the reals as "not complex" instead of "subset of complex" unless you carefully specify otherwise and have a good reason to do so.

This entire thread is basically validating the accuracy of the guess you made in the initial post... The answer is indeed "nothing"
 
  • #14
Fair enough.
 
  • #15
I think the observable associated with that matrix is how many lives Schroedinger's cat has.
 
  • #16
Since the 12 x 12 multiplication table has no significance in the context of Hermition operators (the OP's guess), I am closing this thread.
 

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