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

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Discussion Overview

The discussion revolves around the definition of a matrix and whether its entries can be elements of an unordered set without defined operations. Participants explore the implications of having entries from such sets, contrasting them with traditional definitions involving fields or rings.

Discussion Character

  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants note that traditional definitions of matrices involve entries from a field, while questioning the validity of matrices with entries from unordered sets.
  • It is suggested that while one can still refer to an array of objects as a matrix, the usual matrix operations would not apply if the entries are not from a field or commutative ring.
  • Participants discuss the necessity of having defined operations for matrix addition and multiplication to classify an array as a matrix.
  • One participant proposes that entries of a matrix should at least belong to a semiring, seeking agreement on this assertion.

Areas of Agreement / Disagreement

There is no consensus on whether entries of a matrix can be from an unordered set, and participants express differing views on the necessary conditions for classifying an array as a matrix.

Contextual Notes

Participants highlight the importance of defined operations for matrix classification, but the discussion remains open regarding the specific requirements for entries beyond fields or rings.

CSteiner
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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?
 
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CSteiner said:
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 [itex]nm[/itex]-tuple of objects drawn from some set [itex]X[/itex] 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 [itex]X[/itex] is not a field.
 
So essentialy you're saying it is still fine to call it a matrix, but the usual matrix operations are not defined on it?
 
CSteiner said:
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.
 
micromass said:
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?
 
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.
 
HallsofIvy said:
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!
 
HallsofIvy said:
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?
 

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