# 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. ### 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...

12. ### 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. ### 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

15. ### Does the adjoint of an outer product equal to itself?

So (|a><a|)† = |a><a|
16. ### 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...
17. ### 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...
18. ### 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...
19. ### 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...
20. ### 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...
21. ### 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...
22. ### 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..??
23. ### 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?
24. ### 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...
25. ### 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.