Can the entries of a Matrix be elements of an unordered set?

[CSteiner]

Most definitions of a matrix that I have seen involve entries that are elements of a field. What if I have a unorderd set with no operations defined on it, say a set of different colored marbles or a set of historical events. Can I have a matrix whose entries are elements of such a set?

 

[pasmith]

Most definitions of a matrix that I have seen involve entries that are elements of a field. What if I have a unorderd set with no operations defined on it, say a set of different colored marbles or a set of historical events. Can I have a matrix whose entries are elements of such a set?

Fields don't have to be ordered (for example the complex numbers are not).

Fields do have to have addition and multiplication operations defined on them, on which the rules of addition and multiplication of matrices are based.

Of course the concept of an ordered $nm$-tuple of objects drawn from some set $X$ which is indexed by an integer between 1 and n and a second integer between 1 and m rather than by a single integer between 1 and nm makes sense even if $X$ is not a field.

 

[CSteiner]

So essentialy you're saying it is still fine to call it a matrix, but the usual matrix operations are not defined on it?

 

[micromass]

So essentialy you're saying it is still fine to call it a matrix, but the usual matrix operations are not defined on it?

Depends. If your entries are that of a field (or commutative ring in more generality), then you can call it a matrix and it has the usual matrix operations, regardless of whether the field has an order.

 

[CSteiner]

Depends. If your entries are that of a field (or commutative ring in more generality), then you can call it a matrix and it has the usual matrix operations, regardless of whether the field has an order.

But if my entries are not members of a field what do I call it?

 

[HallsofIvy]

An array can have whatever entries you want (presumably with some reason for their position in the array). A matrix, however, must have matrix addition and multiplication defined so you must be able to "multiply" and "add" the individual elements of the matrix.

 

[CSteiner]

An array can have whatever entries you want (presumably with some reason for their position in the array). A matrix, however, must have matrix addition and multiplication defined so you must be able to "multiply" and "add" the individual elements of the matrix.

That clears it up, thanks!

 

[CSteiner]

An array can have whatever entries you want (presumably with some reason for their position in the array). A matrix, however, must have matrix addition and multiplication defined so you must be able to "multiply" and "add" the individual elements of the matrix.

Okay, so I've been doing more reading and I think that based on your criteria we can say that at the very least, entries of a matrix must be members of a semiring. Would you agree with this statement?