The Multiplication Table is a Hermitian Matrix

[jaurandt]

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...

 

[vanhees71]

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.

 

[Haborix]

$M_{ij}=ij$, where, in this case, $i,j=1,\cdots,12$.

 

[fresh_42]

$M_{ij}=ij$, where, in this case, $i,j=1,\cdots,12$.

What did you multiply?

 

[Haborix]

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.

 

[fresh_42]

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?

 

[Haborix]

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.

 

[jaurandt]

To specify, yes a 12 by 12 matrix where element_ij = i*j where elements_ii = 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.

 

[Mark44]

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.

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.

 

[Nugatory]

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.

 

[jaurandt]

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.

 

[Mark44]

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?

 

[Nugatory]

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"

 

[jaurandt]

Fair enough.

 

[PeroK]

I think the observable associated with that matrix is how many lives Schroedinger's cat has.

 

[Mark44]

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.