What is Outer product: Definition and 25 Discussions
In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions n and m, then their outer product is an n × m matrix. More generally, given two tensors (multidimensional arrays of numbers), their outer product is a tensor. The outer product of tensors is also referred to as their tensor product, and can be used to define the tensor algebra.
The outer product contrasts with:
The dot product (also known as the "inner product"), which takes a pair of coordinate vectors as input and produces a scalar
The Kronecker product, which takes a pair of matrices as input and produces a block matrix
Standard matrix multiplication
Quantum states are most often described by the wavefunction ,##\Psi##. Variable ,##\Psi(x_1x_2\dots x_n) \Psi^*(x_1x_2\dots x_n)## defines probability density function of the system.
Quantum states can also be described by the density matrices (operators). For a pure state, density matrix is...
Inner product is a generalization of the dot product on spaces other than Euclidean and for vectors it is defined in the same way as the dot product. If we have two vectors $v$ and $w$, than their inner product is: $$\langle v|w\rangle = v_1w_1 + v_2w_2 + ...+v_nw_n $$
where $v_1,w_1...
I will post the answer here, part of which I do not follow.
I do not follow the outer-product part. I know that I should multiply two terms together if they are in the same space. However, in this problem, I do not know how to determin which term belongs to which space. It seems, sometimes...
I have a simple question...the control qubit is A and the the target is B.
The cnot is applied on |1A> <0A|⊗|0B0C>.
...
How does it work.
Thanks in advance.
Hi, I'm trying to understand an outer product |1>_a<1| where |1>_a is the ket for one qubit (a) and <1| is the bra for another qubit. Does this make sense and is it possible to express it in terms of tensor products or pauli matrices?
Hello all. I'm using Griffiths' Introduction to Quantum Mechanics (3rd ed., 2018), and have come across what, on the face of it, seems a fairly straightforward principle, but which I cannot justify to myself. It is used, tacitly, in the first equation in the following worked example:
The...
Reading the Wikipedia entry about the Navier–Stokes equation, and I don't understand this second term, the one with the outer product of the flow velocities. I mean, I understand the literal mathematical meaning, but I don't have an intuitive idea of what it physically represents. When I make...
Hello! I am reading so very introductory stuff on geometric algebra and at a point the author says that, as a rule for calculation geometric products, we have that ##e_{12..n}=e_1\wedge e_2 \wedge ...\wedge e_n = e_1e_2...e_n##, with ##e_i## the orthonormal basis of an n-dimensional space, and I...
Say, we have two Hilbert spaces ##U## and ##V## and their duals ##U^*, V^*##.
Then, we say, ##u\otimes v~ \epsilon~ U\otimes V##, where ##'\otimes'## is defined as the tensor product of the two spaces, ##U\times V \rightarrow U\otimes V##.
In Dirac's Bra-Ket notation, this is written as...
I am trying to learn Geometric Algebra from the textbook by Doran and Lasenby.
They claim in chapter 4 that the geometric product ab between two vectors a and b is defined according to the axioms
i) associativity: (ab)c = a(bc) = abc
ii) distributive over addition: a(b+c) = ab+ac
iii) The...
If one has two single-particle Hilbert spaces ##\mathcal{H}_{1}## and ##\mathcal{H}_{2}##, such that their tensor product ##\mathcal{H}_{1}\otimes\mathcal{H}_{2}## yields a two-particle Hilbert space in which the state vectors are defined as $$\lvert\psi ,\phi\rangle...
Homework Statement
In Sakurai's Modern Physics, the author says, "... consider an outer product acting on a ket: (1.2.32). Because of the associative axiom, we can regard this equally well as as (1.2.33), where \left<\alpha|\gamma\right> is just a number. Thus the outer product acting on a ket...
In my QM textbook, there's an equation written as:
\vec{J} = \vec{L}\otimes\vec{1} + \vec{S}\otimes\vec{1}
referring to angular momentum operators (where \vec{1} is the identity operator). I don't really understand what the outer product (which I'm assuming is what the symbol \otimes means...
Hi. I'm wondering if anyone can point me to any information on techniques to decompose a matrix (actually a 3D matrix) into an outer product of vectors. Particularly, given M_{i,j,k}, I want to find vectors a_{i}, b_{i} and c_{i} such that
M_{i,j,k} = a_{i}b_{i}c_{i}
where the...
Hello,
it is known that given two column vectors u and v with N entries, the quantity uvT is a NxN matrix, and such operation is usually called outer product (or tensor product).
My questions are:
1) How can we test whether or not a matrix M is expressible as an outer product of two...
Homework Statement
Given u, v \in \mathbb{R}^{n}, and A \in \mathbb{R}^{n \times n}, \mathrm{det}\left(A\right) \neq 0, find \mathrm{det}\left( A + uv^{T} \right)Homework Equations
Generic determinant and eigenvalue equations, I suppose.The Attempt at a Solution
Hoping to gain some insight, I...
Show that, ina coordinate basis, any (2,1) tensor T at p can be written as
T=T^{\mu \nu}{}_\rho \left( \frac{\partial}{\partial x^\mu} \right)_p \otimes \left( \frac{\partial}{\partial x^\nu} \right)_p \otimes \left( dx^\rho \right)_p
I have no idea how to start this - any ideas?And secondly...
Hi,
Can someone explain to me how to write a matrix as a sum of outer products like \left|\psi\rangle\langle\psi\right|?
For example how would I do a CNOT gate? http://en.wikipedia.org/wiki/Controlled_NOT_gate
I assume this is fairly easy since it is always assumed and I have kind of picked...
why |a><b| expresses the projection...how can it be possible on matrix..if we multiply a ket a with a bra b ...we get a product of two matrix(one is a column matrix,an0ther is row matrix)..from where nothing can be realized very clearly..how this multiplication of matrix can give a projection..??
Given \left| v\right> and \left| u\right> what is the difference between the outer product \left| v\right>\left< u\right| and the tensor product \left| v\right>\otimes\left|u\right>? Is the latter a matrix representation of the former in some basis? Which basis would that be?
A question arose to me while reading the first chapter of Sakurai's Modern Quantum Mechanics. Given a Hilbert space, is the outer product \mathcal{H}\times \mathcal{H}^\ast \to End(\mathcal{H}); (| \alpha\rangle,\langle \beta|)\mapsto | \alpha\rangle\langle \beta| a surjection? Ie, can any...