Definition talk:Unit Matrix

I don't see why Def 2 has to be removed. It appears to me as a familiar way of describing a concept both in a concrete and an abstract way. --Lord_Farin (talk) 08:24, 30 January 2013 (UTC)


 * Does a matrix over an arbitrary ring actually have an identity? If the ring has no unity can there be such an identity? My intuition may be letting me down here, but I would have thought that it can't happen. --prime mover (talk) 08:26, 30 January 2013 (UTC)


 * The identity matrix (EITHER definition) only exists in a ring with unity. --Dfeuer (talk) 08:28, 30 January 2013 (UTC)


 * To clarify, the Kronecker delta doesn't mean anything if there is no $1$. --Dfeuer (talk) 08:29, 30 January 2013 (UTC)


 * You are right; in e.g. matrices over $2 \Z$ there is no identity matrix. This does only imply that Def 2 needs to be expanded to make sure of this. A reference to a (presumably as of yet unwritten) page like Ring of Matrices has Unity iff Underlying Ring has Unity or something avoiding "Underlying Ring". --Lord_Farin (talk) 08:30, 30 January 2013 (UTC)


 * it seems a bit backwards to me: defining the identity matrix as the matrix which is the unity of the ring of nxn matrices. A priori that's what it is - the fact that Def 1 specifies what that matrix is makes it considerably stronger.


 * If we had a proof which specifically constructed the identity matrix, given its property as in def 2, and made it clear that this is the only matrix that can have these properties, that would perhaps be more interesting - but as it is, the pag~e linking the two defs together merely expresses the fact that the two are the same, and hides the salient facts underneath its technical detail. (Yes I know that's my fault for writing a substandard page in the first place, that's not the point.)


 * Also, the notation's inconsistent and might do with being rationalised. --prime mover (talk) 08:40, 30 January 2013 (UTC)


 * The uniqueness of course follows from the fact that a ring can only have one unity. I'm not sure if there's a way to construct it backwards, but it does seem to me that def2 does define the object, and expresses its most important property, so I think it fits the multiple transcluded definitions model well. I don't think it should be def1, but def2 seems a good spot. --Dfeuer (talk) 08:46, 30 January 2013 (UTC)

Construction is in fact quite possible, from the fact that $I_2 I_1 = I_1$, where $I_2$ is the identity (def 2) and $I_1$ is the identity (def 1). --Dfeuer (talk) 08:53, 30 January 2013 (UTC)


 * But that presupposes the existence of the identity matrix as given in def 1: "We have this object: look, we can construct an object to have the properties as the identity: ooh look, it's exactly the same as this (pre-existing) object that we define it on." --prime mover (talk) 08:57, 30 January 2013 (UTC)


 * Using a general diagonal matrix instead of $I_1$ proves that no identity matrix (in the def2 sense) can exist if $R$ has no unity. --Dfeuer (talk) 09:01, 30 January 2013 (UTC)


 * That's as may be (and needs a page to demonstrate it, well volunteered that man), but what I'm saying is: given the ring of nxn square matrices over a ring with unity, we could do with a page which constructs, from first principles, the matrix that is the identity, rather than saying: "look, this matrix with units down its diagonal behaves like an identity", because this presupposes that someone somewhere has already created this matrix. We need to write a page which generates this matrix. --prime mover (talk) 09:11, 30 January 2013 (UTC)


 * I don't see that as being as necessary as you see it, but I'm not opposed to the concept. --Dfeuer (talk) 09:15, 30 January 2013 (UTC)


 * I consider it as being tantamountly necessary. How was the identity matrix defined in the first place? Guesswork? --prime mover (talk) 09:16, 30 January 2013 (UTC)