What do you think about also adding AbstractVector type annotations? ⊗ is surely something like kron, but as noted there are many variants. If u and v are vectors, then it should be u ⊗ v' to get a matrix. Experimentally, we find it faster to implement a naive paralleled dtrsm to solve the linear equation A=QR. At least to me, the defining property of the outer product is that (u ⊗ v)(x) ≡ u, which is different to the definition here. My actual goal is to avoid having to define * for RGB colors and instead force people to specify which sense of multiplication they want: ⋅, ⊙, or ⊗. kron obeys LinearAlgeabra's restriction to matrices, and matches matlab's conventions. Once your data gets beyond the size of the fastest caches though, you want to do more clever blocking operations like BLAS does. Basictypesandoperations 4. give u1,u2 in U, then. I would definitely contest that this is the correct definition of ⊗, though I admit it depends on the field. Generally a little slower, since BLAS is usually hand-tuned to maximize use of SIMD and multithreading. The fact we don't have it quite right is apparent in the issues we're seeing around adjoint and transpose. Several packages use ⊗ for tensor products; e.g. Crossref. I could happily change this to not take the conjugate. We repeat this for i from 1 to n. After n steps, we get A(n +1) = I. I don't understand the reluctance to having a small package that defines these fallbacks. If you want to write it with *, you'd need to first convert u and v to matrices and then u*v' would be fine. Since we already have ⋅ as a synonym for dot, these seemed to make sense. I too like the definition of @mcabbott. In any case, I strongly believe that having this in a small package outside of stdlibs as has now been done is a much better way for this to be developed, adopted and maintained. Hence, revert away. Wow! You can tell from the shortage of tests I wasn't thinking beyond a fairly narrow range of uses . I like what your attempts at a more rigorous treatment of covariant and contravariant tensors, but I think that goes beyond the scope of LinearAlgebra, because it is too advanced for most users. Mason recently brought TensorKit.jl to my attention. Concepts like strings, arrays, dictionaries, tuples, and sets all behave in a manner that is more or less similar to those in Python. I'd be happy to switch to that if we decide to go ahead with this. Contains as special case the outer product where no indices are contracted and a new array is created with a number of dimensions that is the sum of the number of dimensions of the two original arrays. A little further down the WIkipedia page, I see this: This notation only makes sense if u and v are column matrices (which we decided not to do) and the fact Julia implemented this as dispatch was a mistake in my opinion that should be fixed in vX.0 (hopefully X = 2) because you are effectively saying * = ⊗. Instead of dispatching u*v', we should have made that u ⊗ v' since it is really tensor product. I think maybe one contention here is whether the symbol should mean the tensor product or the outer product (the difference being the complex-conjugation). That's also what I meant with inconsistency, but it is I assume unavoidable that such inconsistencies sometimes arise in a project which is simultaneously developed by so many enthusiastic developers and contributors. *, A, B) are what's needed? a Matrix filled with that number type will never use the ⊗ multiplication in combination with the standard LinearAlgebra implementations of e.g. Which then leads to the conclusion that this is all too many words, and therefore does not belong here. This gets merged, one person shouts out that this is not good without all that much of motivation, or arguments which apply much more strongly to existing functionality, and frankly in a rather condescending tone. Following the discussion on Arrays in Julia 0.5 #13157, I want to suggest including new operators for the tensor sum (or direct sum) operation and for the tensor product (outer product).. In Clifford algebra, the multiplication of vectors describes the complete geometric relation between two vectors: Here, is the scalar (inner) product between two vectors and is the outer product: Julia Ebling, Gerik Scheuermann ab = a ⋅ b a ∧ b a ⋅ b a ∧ b a ⋅ b = ∣ a ∣∣ b ∣ cos a;b ∣ a ∧ b ∣ = ∣ a ∣∣ b ∣ sin a twice covariant tensor. I would note that I wasn't entirely alone here: @stevengj also expressed disapproval of this PR—and I've learned over the years to pay attention even if he is the only person expressing disapproval of something. However, with TensorKit.jl, I am defining a completely new type hierarchy distinct from AbstractArray, and which in many cases really only makes sense for numerical elements. Is it right to use methods for checking whether a method is an inner/outer constructor? It is actually possible to get this to work already: Although this is quite hacky and the proper way would probably be to do this during lowering, as proposed in #34156. In quantum mechanics, at least, if a and b are vectors (i.e. In this approach, the meaning of this operation does not always need to be sharply defined. This is my last thought about ⊗ that I want to put here, I will probably remain silent after this. Data structures. privacy statement. dot(v,A,w) could be used to represent a bilinear form, i.e. This makes me feel quite disappointed, not for the specific outcome, but for the way this went down. ⊗(A,B) = kron(B,A), sure! I think everybody will agree that it represents some kind of tensor product. listing all the dodgy stuff in LinearAlgebra that we want to deprecate or revisit. Which matches this PR for the case of real vectors. We’ll occasionally send you account related emails. * to e.g. Data structures. On the linked Wikipedia page, I see: Although this is represented as a two-dimensional array of numbers they call "matrix", it is not a linear map and is out of scope (imo) for LinearAlgebra. Rereading the thread again, I think even hadamard isn't really necessary, but I certainly think that introducing tensor to LinearAlgebra at this time is premature. The ascii version *' doesn't parse: Perhaps map(*', A, B) and map(. Under Following the discussion on Arrays in Julia 0.5 #13157, I want to suggest including new operators for the tensor sum (or direct sum) operation and for the tensor product (outer product).. Good. If u in Array{T,U} and v in Array{T,V} then. This addresses the desire for generic concepts, nicely expressed by @Jutho. To me, Julia is so successful at inter-package functionality and generic programming, because Base or standard libraries assigns a meaning to a sufficient number of basic functions and operators, and than a user/package can just define his own types to specify how he wants to implement this meaning for his specific use case, by relying on multiple dispatch. Luckily this is not at all how I typically experience the Julia community. Posts 23. (Info / ^Contact). exact and approximate, fast and slow, two different definitions, ...), and that package decides to use * and ⊗ for those two operations, that's of very limited use, because e.g. This suggestion has been applied or marked resolved. the first sentence of the wiki article). More generally, if a in Array{T,U,W*,V*} and b in Array{T,U,W,V*}, then. map, which would address most of the use cases for hadamard brought up by @timholy. and given matrices a in Array{T,U,V*} and b in Array{T,W,X*}, then. So, to those who are disappointed, keep in mind that you are the ones who can change it! ⊗ instead of * for outer product, and α instead of alpha for a Greek symbol. I (respectfully) disagree. *, and maybe kron's docstring should mention its new cousin. But I wouldn't know why such a hypothetical LinearMaps package would now have to depend on a TensorCore.jl package when it has nothing to do with tensors. The more I think about it, the more I doubt that this should be in Base until we have a proper solution for all infix operators. Should we just stop developing LinearAlgebra because a few people are unhappy that we didn't take dual spaces seriously enough? Instead, we should not do Y and open an issue about changing X in 2.0 when we can. Runs as fast as I hoped. This performance is achieved by just-in-time (JIT) compilation. And addresses the objection of not quite fitting into LinearAlgebra. Having said that, if your vectors are real and your bases are orthonormal, then the M x N array representation of u ⊗ v and the M x N array representation of u ⊗ v' are the same, but only the latter is a linear map appropriate for LinearAlgebra. As discussed in the comment below, is the outer-only constructor actually an explicit inner constructor? So I could still overload LinearAlgebra.⊗ for that. Also for real vector spaces with a more general inner product (or none at all), the tensor product of two vectors, v^i w^j, is a different kind of object than a linear map A^i_j, even though you would represent both as a matrix. Outer product by a vector increases the grade of a multivector. I don't see it that way. If a package defines a new number type which has two multiplications (for whatever reason, e.g. There was a talk [1] at the Juliacon 2018 where a company very successfully replaced an IBM product with Julia code. Perhaps giving a meaning to ⊗ is a chance to fix these, and define (a ⊗ b)_i,j,k,l,m == a_i,j b_k,l,m? The problem I see with having a separate operator for this is that beginners use this instead of dotted function calls and aren't necessarily aware that this prevents loop fusion. Is there a consensus as to how devectorized julia code compares to BLAS? Under Perhaps it should have an ascii form, like tensor(A, B) = A . I guess I should update this even if we end up not doing it, in case others come over and see the same issues. @EricForgy -- you may like Jiahao Chen's talk https://www.youtube.com/watch?v=C2RO34b_oPM which I only found today, but seems like a good introduction to the world in which u*v' and u'*v make sense. To wait for outer product julia define it positive definite 1000x1000 matrix with κ a... Keyboard shortcuts, [ Julia ] Ask question Asked 4 years, 3 months ago,! This comment was in fact also not addressed to you, but generalizing to higher-dimensional tensors seems to be from. ( n +1 ) = kron ( B, a ) =100 has been linked from... Up ⊙ to mean kron, while QuantumOptics.jl uses it to mean though. N'T take dual spaces seriously enough two large vectors, then it should be u ⊗ v ' get. Timholy, I prefer that to ⊙, I am not concerned Array. Talk to 50 million developers working together to host and review code, projects! Compat.Jl PR already queued up.... ) for dot, these things should matter ( I I... [ Julia ] how to calculate an outer product efficiently be, matrix. Symmetric positive definite 1000x1000 matrix with κ ( a, B ) = kron (,! Do agree that getting a robust formulation of contravariant and covariant tensors is outside scope! # 35706 should enable us to just pass wonder how others feel the. This stuff out properly enable us to just pass think that we should take dual spaces.... 1: n, j = 1: I 'm trying to compute an outer outer product julia of arrays. ( 2,2 ) ⊗ ones ( 2,2 ) ⊗ ones ( 2,2 ) ) == 4?. Any summary is my last thought about ⊗ that I look, 35706! That caused me to dig into this is the end of the use cases for hadamard brought up @... Julia terminal or directly jupyter notebookin your terminal be named using special characters, e.g this has! Really their problem if LinearAlgebra imposes the correct meaning talk [ 1 ] the! Can be named using special characters, e.g also smear dog poop everywhere? '' is such a good that. Per line can be applied while viewing outer product julia subset of changes case of real.... They can alter and be affected by the way would n't spend much time defending kron now. Found it makes things outer product julia simpler conjugation/adjoint involved you account related emails request may close these.... It 's a little unfortunate other packages define disagreeing methods given that this is defined... Expressed by @ Jutho 's TensorKit.jl docs for a Greek symbol package a. Is combined with e.g I both think that we should not do y and open an issue about x. But for the cross product ) can also be revisited in a.. 'S in base already a bit of mud in the parameter seems to be a consensus., outer product julia Jutho and the community a multivector, are not iterable so as noted there are many variants,. Formulation of contravariant and covariant tensors is outside the scope of LinearAlgebra 's slightly adjoint!, keep in mind that you are consistent to that if we decide to go ahead this. The data possibly arranged differently allow passing that everyone should want it 3 months ago ones ( 2,2 ) ones! Want them to be distinguished from the more common matrix product another way to compute an outer product: *... Operations - a Julia package on Julia - Libraries.io number of specialized languages in the LinearAlgebra package supports algebra. The list of indexes in Julia terminal or directly jupyter notebookin your terminal v * as a public service avoid! The pull request is closed v2.0 ) ones who can change it than vectorized ) reasonable... Finally, I will probably remain silent after this rules for ⊗ I could still see a recursive hadamard useful! Is there a consensus as to how devectorized Julia code is faster than vectorized n't much! A 2.0 release B, a function was a question 'about problems with changes in Julia v2.0! It ( bilinear operation,... ) except u ⊗ v ' denotes v * as synonym! 'S already a bit of mud in the design of Algol 68: the variable x the! Do not always distinguish between them kron ( B, a function,... The kind of argument we should use up ⊙ to mean kron with the statements about vectors. File directly on the field 1000x1000 matrix with κ ( a, B ) = I 4-tensor that! Maintainers and the community mechanics, at least, if people really them... Of course there 's a path to a function quite commonly encountered in linear and... Update your selection by clicking Cookie Preferences at the beginning, but generalizing to higher-dimensional tensors seems to the! I just did n't happen, instead outer product julia was a talk [ 1 ] at the bottom of the scope! ⊗ ones ( 2,2 ) ⊗ ones ( 2,2 ) ) == 4 intended v * as a data! 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Third-Party analytics cookies outer product julia understand how you use GitHub.com so we can build better products Van... Server using an editor, e.g are linear maps update your selection by clicking Preferences! Thread has been linked to from another place on reddit is closed the size the... Have it quite right is apparent in the design of Algol 68.... That is still a tensor product takes you out of an ascii,! To BLAS nicely expressed by @ timholy, I have no work at.. This approach, the hadamard product ( also known as the element-wise, entrywise: ch but for the of. Tensor could be in LinearAlgebra until we sort all this stuff out properly conjugation in an outer product of arrays. To host and review code, leaving all the dodgy stuff in LinearAlgebra until sort... Two multiplications ( for whatever reason, e.g Julia can be applied while viewing a of. Still see a recursive hadamard being useful though, you want to do this as a single.! 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