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

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  1. Dario56

    I Density Operators of Pure States

    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...
  2. Dario56

    I Inner and Outer Product of the Wavefunctions

    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...
  3. Haorong Wu

    How to calculate the outer product in GR?

    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...
  4. G

    A The cnot with outer product control qubit

    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.
  5. S

    Understand the Outer Product of two qubits

    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?
  6. P

    I Integration of the Outer Product of a Basis

    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...
  7. snoopies622

    I Outer product of flow velocities in Navier-Stokes equation

    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...
  8. S

    I Outer product in geometric algebra

    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...
  9. D

    A Difference Between Outer and Tensor

    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...
  10. malawi_glenn

    I Is Geometric Algebra inconsistent/circular?

    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...
  11. F

    I What is the outer product of a tensor product of vectors?

    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...
  12. L

    Hermitian conjugate of outer product

    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...
  13. kq6up

    Is the trace of an outer product always equal to 1?

    Is the trace of an outer product of a normalized state eq. (psi) equal to 1? Thanks, Chris Maness
  14. Sudharaka

    MHB Difference Between Tensor Product and Outer Product

    Hi everyone, :) Xristos Lymperopoulos on Facebook writes (>>link<<);
  15. D

    How Does the Outer Product Operate on Quantum Mechanical Operators?

    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...
  16. P

    Decompose matrix into outer product of vectors

    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...
  17. mnb96

    Expressing matrices as outer product of two vectors

    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...
  18. S

    Determinant of outer product matrices

    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...
  19. L

    Efinition of the outer product tensor

    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...
  20. R

    Writing a matrix as an outer product expansion.

    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...
  21. H

    Outer product problem of derac notation

    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..??
  22. D

    Difference between the outer product

    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?
  23. J

    Outer product in Hilbert space

    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...
  24. R

    Outer product of two one forms.

    Given two one forms f = (1,1,0,0,) and g=(-1,0,1,0): what are the components of f(x)g ... would appreciate any help.