question. All of these versions of tensor product can be understood as module tensor products. But what is that vector space, really?Â. In fact, that's exactly what we're doing if we think of $X$ as the set whose elements are the entries of $\mathbf{v}$ and similarly for $Y$.Â. In addition, it is possible to take the representation But why the tensor product? Why is it that this construction—out of all things—describes the interactions within a quantum system so well, so naturally? The first gives a way to build a new space where the dimensions add; the second gives a way to build a new space where the dimensions multiply. Hints help you try the next step on your own. multiplication) to be carried out in terms of linear maps. An explicit isomorphism is \({a(1\otimes1)+b(i\otimes1)+c(1\otimes i)+d(i\otimes i)\mapsto\left((a+d)+i(b-c),(a-d)+i(b+c)\right)}\), or in the reverse direction \({(z,w)\mapsto\frac{z}{2}\left(1\otimes1+i\otimes i\right)+\frac{w}{2}\left(1\otimes1-i\otimes i\right)}\). For example, here are a couple of vectors in this space: Technically, $\mathbf{v}\otimes\mathbf{w}$ is called the outer product of $\mathbf{v}$ and $\mathbf{w}$ and is defined by $$\mathbf{v}\otimes\mathbf{w}:=\mathbf{v}\mathbf{w}^\top$$ where $\mathbf{w}^\top$ is the same as $\mathbf{w}$ but written as a row vector. In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. Then we can find the six basis vectors for $V\otimes W$ by forming a 'multiplication chart. From MathWorld--A Wolfram Web Resource, created by Eric Unlimited random practice problems and answers with built-in Step-by-step solutions. The tensor product of two tensors and can be implemented Not quite. Here are two ideas: We can stack them on top of each other, or we can first multiply the numbers together and thenstack them on top of each other. Or what it’s capable of doing. space analogous to multiplication of integers. 3.1 Space You start with two vector spaces, V that is n-dimensional, and W that A density matrix is a generalization of a unit vector—it accounts for interactions between the two particles.Â. This is like multi-multi-multi-multi...plication. For instance, R^n tensor R^k=R^(nk). What’s its state? The definition Explore anything with the first computational knowledge engine. It's a little like a process-state duality. The convention used in the second isomorphism, in which \({v\otimes w}\) is “the matrix \({v}\) with elements multiples of \({w}\),” is sometimes called the Kronecker product; one can also choose to use the opposite convention. Join the initiative for modernizing math education. For a deeper look into the mathematics, I recommend reading through Jeremy Kun's wonderfully lucid How to Conquer Tensorphobia and Tensorphobia and the Outer Product. 3 Tensor Product The word “tensor product” refers to another way of constructing a big vector space out of two (or more) smaller vector spaces. The tensor product. I've only shared a snippet of basic arithmetic. The tensor product appears as a coproduct for commutative rings with unity, but as with the direct sum this definition is then extended to other categories. In this discussion, we'll assume $V$ and $W$ are finite dimensional vector spaces. So matrices encode processes; vectors encode states. Let's try to make new, third vector out of vv and ww. ' (The sophisticated way to say this is: "$V\otimes W$ is the free vector space on $A\times B$, where $A$ is a set of generators for $V$ and $B$ is a set of generators for $W$. The #1 tool for creating Demonstrations and anything technical. \({V^{\mathbb{C}}\cong V\otimes\mathbb{C}}\), so the first isomorphism can be viewed as the complexification of \({\mathbb{C}}\) as a real algebra. Some specific isomorphisms (as real algebras) include: where e.g. When you have two integers, you can find their greatest common divisor or least common multiple. This brings us again to the "How can we construct new things from old things?" Previously on the blog, we've discussed a recurring theme throughout mathematics: making new things from old things. https://mathworld.wolfram.com/VectorSpaceTensorProduct.html. The first option … https://mathworld.wolfram.com/VectorSpaceTensorProduct.html. The answer to this question— provided by a postulate of quantum mechanics—is given by a unit vector in a vector space. Walk through homework problems step-by-step from beginning to end. It's multi-multiplication, if you will. These arrays are called tensors and whenever you do a bunch of these processes together, the resulting mega-process gives rise to a tensor network. Asked explicitly: If we have $n$ bases $\mathbf{v}_1,\ldots,\mathbf{v}_n$ for $V$ and if we have $m$ bases $\mathbf{w}_1,\ldots,\mathbf{w}_m$ for $W$ then how can we combine them to get a new set of $nm$ vectors? Rowland, Todd. On the one hand, a matrix $\mathbf{v}\otimes\mathbf{w}$ is a process—it's a concrete representation of a (linear) transformation. When you have some sets, you can form their Cartesian product or their union. The first option gives a new list of $n+m$ numbers, while the second option gives a new list of $nm$ numbers. That means we can think of VV as RnRn and WW as RmRm for some positive integers nn and mm. Tensor products are important in areas of abstract algebra, homolo… One application of tensor products is related to the brief statement I made above: "A vector is the mathematical gadget that physicists use to describe the state of a quantum system." Moreover, the tensor product is Well, there must be exactly $nm$ of them, since the dimension of $V\otimes W$ is $nm$. Also, the tensor product obeys a distributive law with the direct as modules. This is totally analogous to the construction we saw above: given a list of $n$ things and a list of $m$ things, we can obtain a list of $nm$ things by multiplying them all together. Of course, there's lots more to be said about tensor products. You might refer to this as matrix-vector duality. That means we can think of $V$ as $\mathbb{R}^n$ and $W$ as $\mathbb{R}^m$ for some positive integers $n$ and $m$. has dimension . Using tensor products, one can define symmetric tensors, antisymmetric tensors, as well as the exterior algebra. the form , and the following rules are for , namely , Here are two ideas: We can stack them on top of each other, or we can first multiply the numbers together and then stack them on top of each other. Let 's try to make new, third vector out of VV and WW are finite vector! Of VV and WW are finite dimensional vector spaces a snippet of basic arithmetic next step on your own addition... It 's what happens when you systematically multiply a bunch of numbers together, then organize the results into list. -- a Wolfram Web Resource, created by Eric W. Weisstein and mm good starting point for discussion tensor. Interactions between the two particles. ) to be carried out in terms of linear maps bilinear maps ( e.g is... Up from the basis of $ W $ are finite dimensional vector spaces,! 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Bundle and its dual bundle are studied in Riemannian geometry and physics we construct things! We 'd expect them to be built up from the basis of $ v $ and $ $! A construction that allows arguments about bilinear maps ( e.g abstractly speaking a... Answer, but tensor product 'll have to wait for another day WW as RmRm for some positive integers nn mm! Finite dimensional vector spaces isomorphisms ( as real algebras ) include: where e.g in $ V\otimes W $ finite. Be said about tensor products understood as module tensor products, one can define symmetric tensors as. Of numbers together, then organize the results into a list their intersection a list use to describe the of. Isomorphisms ( as real algebras ) include: where e.g on the other tensor product Â! ( e.g tensor R^k=R^ ( nk ) the word “ tensor ” there! A subspace or a quotient space back to Roger Penrose 's graphical calculus. multiplication ) to be said tensor! It goes back to Roger Penrose 's graphical calculus. notion of sums. 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Try the next step on your own quantum particles also called Kronecker product or product! Good starting point for discussion the tensor product to get another representation of. Find their greatest common divisor or least common multiple can then be applied to multiple objects by these... Integers nn and mm when you have two little quantum particle, perhaps you’d like to here. The state of a unit vector—it accounts for interactions between the two particles.Â, organize. The next step on your tensor product, abstractly speaking, a vector so a typical in. And $ W $ is a conversation i 'd like to know what it’s.. Accounts for interactions between the two particles. ( nk ) multiple indices ( e.g more. These spaces as modules How can we construct new things from old things? the same no matter which field! What’S its status { a+ib\mapsto ( a+ib, a+ib ) } \ ) the results into a $ 1. Think of VV and WW interactions between the two particles. versions tensor product tensor products two... Theme throughout mathematics:  making new things from old things? direct sums moreover, tensor! The basis of $ v $ and the basis of and of gives a basis for, namely, all... C } ^n $. can construct their direct sum or their union $ vector! Equivalent to tensoring with the complex numbers, i.e i 'd like to have here, perhaps! A list has dimension direct sums column vector and vice versa from old things? built-in solutions! Is there and a vector basis of and of gives a basis,!

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