Wikipedia:Reference desk/Archives/Mathematics/2014 July 26

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July 26

Type conversion

Is there any standard mathematical notation for specifying the type (e.g. scalar, vector, matrix) and dimensions of an otherwise ambiguous expression? For example, can the zero matrix of some unknown dimensions x×y and the scalar zero be represented by separate symbols that are standard and unambiguous, given that 0 can mean either? (Intuitively I'd think $\left(0:0\in \mathbb {R} \right)$ and $\left(0:0\in \mathbb {R} ^{x\times y}\right)$ would be comprehensible, but possibly more awkward than necessary.) Also, is it possible to distinguish the empty set whose sum is scalar zero from the empty set whose sum is a zero matrix? NeonMerlin 13:00, 26 July 2014 (UTC)

Computer programmers distinguish, but mathematicians do not. The zero scalar is identified with the zero matrix, and no distinction is called for. Bo Jacoby (talk) 23:02, 26 July 2014 (UTC).

That's completely wrong. --Trovatore (talk) 02:09, 27 July 2014 (UTC)

The Mathematics_of_general_relativity#Energy_conservation has

$T^{ab}{}_{;b}\,=0$

not

$T^{ab}{}_{;b}\,=0^{a}$

Bo Jacoby (talk) 07:26, 27 July 2014 (UTC).

But surely by 0, the RHS means the zero vector, rather than the zero scalar. No identification has occurred. Sławomir Biały (talk) 14:51, 27 July 2014 (UTC)

Exactly. The zero vector $0^{a}$ is written 0. No distinction is called for. Bo Jacoby (talk) 14:58, 27 July 2014 (UTC).

It is clear from the context here that the notation 0 means the zero vector rather than the zero scalar. No one seriously believes there is no difference between these (which is what "is identified with") would mean, and it would be perfectly reasonable to write 0^a if there were any risk of confusion. You would not see the vacuum equation written as $R_{a}^{b}=1/2R$. Sławomir Biały (talk) 15:05, 27 July 2014 (UTC)

Also, if you were to believe that scalars and vectors were naturally identified, then you would write the Einstein vacuum equation as $R_{a}^{b}=1/2R$. As far as I know, no one writes it this way, even though scalars do naturally embed into the endomorphism algebra. Since no one writes it this way, it would be rather mystifying if they regarded the zero on the right hand side of the equation $G_{a}^{b}=0$ as a scalar, rather than the zero endomorphism. And for your example, there isn't a natural embedding of the scalars into the space of contravariant vectors to begin with, so claiming that the zero on the RHS of the equation is the same as the zero scalar is obvious nonsense. Sławomir Biały (talk) 21:47, 27 July 2014 (UTC)

I respectfully disagree. I, for one, seriously believe that "there is no difference between these". $0^{a}-0=0$. Where do I find the vacuum equation? Bo Jacoby (talk) 18:35, 27 July 2014 (UTC).

You may believe that there is no difference between the zero vector and the zero scalar. But you are wrong. It only makes sense to "identify" scalars with vector quantities when there is a natural embedding of the scalars into the vector space (e.g., working over an algebra). The zero vector and zero scalar, in this setting, live in completely different spaces. Do not be confused by the fact that the same symbol is used for these different mathematical objects. And this is not the place to push your crank theory that everything denoted by the symbol 0 is the same thing. That has already been conclusively refuted by others. Sławomir Biały (talk) 19:52, 27 July 2014 (UTC)

Bo, to answer your question: at Einstein field equations#Vacuum field equations. Actually, in the context of tensor component equations, there is room for interpretation: interpreted as a whole bunch of equations on numeric components, the problem does not arise (each component is just a real number); however this interpretation of the notation rapidly loses value, for example when the covariant derivative is used (as in your example), for which the component-wise equations make no real sense. Sławomir's argument is entirely unaffected by this notational thing. If you want to be able to equate a zero scalar with a zero vector, you need to embed them both in the same algebra (which is possible: see Tensor algebra), but in this case you are formally treating them as part of the same algebra, and permitting addition of tensors of differing order. —Quondum 22:28, 27 July 2014 (UTC)

Thanks. The vacuum field equation is written

$R_{\mu \nu }=0\,.$

The zero on the right hand side is to be understood as a zero tensor field. The OP asked: "can the zero matrix of some unknown dimensions x×y and the scalar zero be represented by separate symbols", and my answer is that zero is written 0, no matter if it is the zero scalar or zero vector or zero matrix or zero function or whatever. My examples show that this is correct. I did not mean, (and I hope I did not write), that scalars are generally embedded in vectors. Bo Jacoby (talk) 05:52, 28 July 2014 (UTC).

Ok. But the zero scalar, zero vector, and zero matrix are not identified (your phrasing, that you vigorously defended). They are different mathematical objects that are often denoted by the same symbol.

Also, the Einstein field equations can be written in a number of equivalent ways. I brought it up only because you seemed comfortable enough with relativity to use it as an example. Obviously that is not the case. The vacuum equation can be written as $G_{ab}=0$ or, raising an index with the metric, as $G_{a}^{b}=0$, or. using the definition of the Einstein tensor, as $R_{a}^{b}-1/2R\delta _{a}^{b}=0$, or as $R_{a}^{b}=1/2R\delta _{a}^{b}$. By taking traces, one would normally see that the Ricci scalar is zero at this point, but it is still a valid tensor equation. It would not be written as $R_{a}^{b}=1/2R$ however. Sławomir Biały (talk) 13:48, 28 July 2014 (UTC)

Thanks. Summarizing: The scalars are not embedded in vectors and so the zero scalar is not the same object as the zero vector even if the two objects share the same symbol 0. The scalars are embedded in (say) 4×4 matrices and the zero matrix is written 0 and the unit matrix $\delta _{a}^{b}$ may be written 1. But an equation like $R_{a}^{b}=1/2R\delta _{a}^{b}$ is not written $R_{a}^{b}=1/2R$ because $\delta _{a}^{a}=4$ is more obvious than trace(1)=4. Do you agree? Bo Jacoby (talk) 06:39, 29 July 2014 (UTC).

Note that $\delta _{a}^{b}$ is not really a tensor or a matrix. It is a tensor element or a matrix element. So the example is not quite relevant. Bo Jacoby (talk) 10:14, 30 July 2014 (UTC).

That would be true if a and b represented particular indices. But they are usually thought of as just placeholders. (Otherwise this actually obviates your original example, too.) This is made precise in the abstract index notation, which we were supposed to have been discussing, in which indexed quantities refer to the underlying tensor, not their components in a coordinate system. Sławomir Biały (talk) 14:53, 30 July 2014 (UTC)

I think I follow Bo, and he's talking of each symbol as an object inhabiting some space (the space being implied by context). So for example, 1 is the identity element of the space that it must be from the context, in his example a space of 2nd order tensors. As an equation in tensor components, $R_{a}^{b}=1/2R$ is not even coordinate-independent, but in the implied 2nd order tensor space, it would be correct. But we do not use this notation because it confuses. In the case of the zero tensor, we do drop the indices, probably because it ends up saying the same thing regarded as an equation in tensor components. I disagree with Bo in the claim that the delta "is not really a tensor or a matrix": it is natural to regard it as either, and it is normal to use it to denote the tensor that is the identity map on vectors (or covectors, though technically one is the transpose of the other). —Quondum 17:15, 30 July 2014 (UTC)

I've seen 1_n and Id_n for the n×n identity matrix. I may have seen 0_x×y, or I may be imagining it. You don't see zero matrices in any notation very often.

Abstract index notation is widely used in relativistic physics. It distinguishes scalars, vectors and matrices (properly rank-2 tensors) by the number of indices, and often their size is implicitly encoded in the letters used for the indices (μν for spacetime indices, ij for spatial indices, etc.). The identity matrix/tensor is sometimes written δ^μ_ν or δⁱ_j and called the Kronecker delta.

I've never seen a notation for a typed empty set, except in programming languages. -- BenRG (talk) 23:29, 26 July 2014 (UTC)

Mathematicians do actually distinguish. There are times when "up to isomorphism" is meant, and no ambiguity results. At other times, two structures can be isomorphic but are intended to be regarded as distinct sets, such as the set of complex numbers being isomorphic to distinct subrings of the quaternions (there is no canonical embedding of C in H, unlike with Z in R). However, it seems pretty normal to "expect" the reader to understand what is "meant", with occasional verbal disambiguation. This is somewhat frustrating to the more literal-minded and to newcomers. The closest seems to be set membership, usually appended as a qualifying statement (e.g. " where x ∈ Q). —Quondum 00:46, 27 July 2014 (UTC)