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DEigensystem
✖
DEigensystem

✖
DEigensystem[ℒ[u[x,y,…]],u,{x,y,…}∈Ω,n]

gives the n smallest magnitude eigenvalues and eigenfunctions for the linear differential operator ℒ over the region Ω.

✖
DEigensystem[eqns,u,t,{x,y,…}∈Ω,n]

gives the eigenvalues and eigenfunctions for solutions u of the time-dependent differential equations eqns.

Details and Options

  • DEigensystem can compute eigenvalues and eigenfunctions for ordinary and partial differential operators with given boundary conditions.
  • DEigensystem gives lists {{λ1,…,λn},{u1,…,un}} of eigenvalues λi and eigenfunctions ui.
  • An eigenvalue and eigenfunction pair {λi,ui} for the differential operator ℒ satisfy ℒ[ui[x,y,…]]==λi ui[x,y,…].
  • Homogeneous DirichletCondition or NeumannValue boundary conditions may be included. Inhomogeneous boundary conditions will be replaced with corresponding homogeneous boundary conditions.
  • When no boundary condition is specified on the boundary ∂Ω, then this is equivalent to specifying a Neumann 0 condition.
  • The equations eqns are specified as in DSolve.
  • N[DEigensystem[…]] calls NDEigensystem for eigensystems that cannot be computed symbolically.
  • The following options can be given:
  • Assumptions $Assumptionsassumptions on parameters
    Method Automaticwhat method to use
  • Eigenfunctions are not automatically normalized. The setting Method->"Normalize" can be used to give normalized eigenfunctions.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Find the 4 smallest eigenvalues and eigenfunctions of the Laplacian operator on [0,π]:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-yzdn77
Out[1]=1

Visualize the eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-wx5bbs
Out[2]=2

Compute the first 6 eigenfunctions for a circular membrane with the edges clamped:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-d3ex
In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-crtyb1
Out[2]=2

Visualize the eigenfunctions:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-q0978
Out[3]=3

Scope  (20)Survey of the scope of standard use cases

1D  (9)

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-dd5aze

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-yjqx5u

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-b0b4f6
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-cvvh62
Out[4]=4

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-ge4exl
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-2d8bfg

Specify homogeneous Neumann boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-kocy8b

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-v0oeps
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-3z82ab
Out[4]=4

This is equivalent:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-ew1oj0
In[6]:=6
✖
https://wolfram.com/xid/0btnz7gc2iv2-snvhso
Out[6]=6

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-cuzc9o

Specify a homogeneous Dirichlet boundary condition:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-c9q6xh

Specify a homogeneous Neumann boundary condition:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-bg0i4l

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-c5axvx
In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-jy4ton
Out[5]=5

Visualize the eigenfunctions:

In[6]:=6
✖
https://wolfram.com/xid/0btnz7gc2iv2-eb28a
Out[6]=6

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-u46ab

Specify a homogeneous Dirichlet boundary condition:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-e05awx

Specify a homogeneous non-zero Neumann boundary condition:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-hm0vt3

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-lya86

The eigenvalues are roots of a transcendental equation:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-nsdqid

Visualize the eigenfunctions:

In[6]:=6
✖
https://wolfram.com/xid/0btnz7gc2iv2-id4r85
Out[6]=6

Specify an Airy operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-cldpj3

Specify a homogeneous Dirichlet boundary condition:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-essb5

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-by1ifr

The eigenvalues are roots of a transcendental equation:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-cnfmi7

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-iqi360
Out[5]=5

Specify an Airy operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-lzzgzu

Specify a homogeneous Neumann boundary condition:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-bzyyfz

Find the 5 smallest eigenvalues and eigenfunctions in an interval:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-di935k

The eigenvalues are roots of a transcendental equation:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-d60btu

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-jy8mb2
Out[5]=5

Find symbolic expressions for the eigenvalues and eigenfunctions of a Laplace operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-cdj4k3

Symbolic eigenvalues:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-gdd6d0
Out[2]=2

Symbolic eigenfunctions:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-j8g1rh
Out[3]=3

Enter the quantum harmonic operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-33s8q5

Find symbolic expressions for the eigenvalues and eigenfunctions on the real line:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-gjib28
Out[2]=2

Specify a heat equation with homogeneous Dirichlet boundary conditions:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-mzmdbz

Find the 4 smallest eigenvalues and eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-1dfazx
Out[2]=2

Visualize the eigenfunctions:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-ribue
Out[3]=3

2D  (6)

Specify a Laplacian operator with homogeneous Dirichlet boundary conditions:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-olvz0

Find the 9 smallest eigenvalues and eigenfunctions in a rectangle:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-q0svtk
In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-yhxj38
Out[3]=3

Visualize the eigenfunctions:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-ey9fmu
Out[4]=4

Specify a Laplacian operator with homogeneous Neumann boundary conditions:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-dzc5oc

Find the 4 smallest eigenvalues and eigenfunctions in a rectangle:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-3cf8zi
In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-qpg6gs
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-snrji
Out[4]=4

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-hz8azd
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-sq6djy

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-tzp4z

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a unit disk:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-646c3s
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-4ir027

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-dvcswe
Out[5]=5

Specify a quantum harmonic oscillator operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-x4itsm

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-mdxmrr

Find the 6 smallest eigenvalues and eigenfunctions of the operator on the plane:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-tdyb7u
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-vmgimx
Out[4]=4

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-wiu9y7
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-calu3w

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-bs5esh

Find the 6 smallest eigenvalues and eigenfunctions of the operator in a triangle:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-bwxx0l
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-l49dkt
Out[4]=4

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-bp7icj
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-hqm92

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-kd8b62

Find the 4 smallest eigenvalues and eigenfunctions of the operator in a sector of a disk:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-fc0694
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-b0jk9w

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-lyqv9v
Out[5]=5

3D  (5)

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-5soch

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-idhpb

Find the 7 smallest eigenvalues and eigenfunctions in a cuboid:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-lqu653
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-beoef7
Out[4]=4

Visualize an eigenfunction:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-coor33
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-d0ilyf

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-i9ulj9

Find the 7 smallest eigenvalues and eigenfunctions in a cylinder:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-jo5954
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-c137sl

Visualize an eigenfunction:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-4wlns
Out[5]=5

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-chbrb7

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-cwisu0

Find the 7 smallest eigenvalues and eigenfunctions in a ball:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-j6wop0

Visualize an eigenfunction:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-bhg57v
Out[4]=4

Specify a Laplacian operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-hhf5pu

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-tdyqt

Find the 7 smallest eigenvalues and eigenfunctions in a prism:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-j4mv75
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-jct22c
Out[4]=4

Visualize an eigenfunction:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-bewipj
Out[5]=5

Specify a quantum harmonic oscillator operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-o98g3b

Specify homogeneous Dirichlet boundary conditions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-8oi6tm

Find the 8 smallest eigenvalues and eigenfunctions throughout space:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-gp270a
In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-evmcm5
Out[4]=4

Visualize the eigenfunctions:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-2klld7
Out[5]=5

Options  (2)Common values & functionality for each option

Assumptions  (1)

Use Assumptions to simplify a result:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-ecanzm
Out[1]=1

Without the option, an equivalent but more complicated answer would be returned:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-yyrzi2
Out[2]=2

Method  (1)

Normalize the eigenfunctions for a differential operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-gq3jd1
Out[1]=1

Verify that the eigenfunctions are normalized to unity:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-bn7w8g
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Compute the first three terms in the eigenfunction expansion of the function with respect to the basis provided by a 1D Laplacian with a Dirichlet condition on the interval :

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-chgfue
Out[1]=1

Compute the Fourier coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-euk8i9
In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-d8ezwq
Out[3]=3

The required eigenfunction expansion is:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-khjz1n
Out[4]=4

Compare the function with its eigenfunction expansion:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-byveyd
Out[5]=5

Build a solution of the heat equation by using a linear combination of eigenfunctions for the heat equation with a Dirichlet condition:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-ee0l91
In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-qsijm
Out[2]=2

Form a linear combination of eigenfunctions:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-q3w8c
Out[3]=3

Verify that this is indeed a solution of the heat equation:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-h2j74j
Out[4]=4

The solution satisfies the homogeneous Dirichlet condition:

In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-bjjefv
Out[5]=5

Visualize the solution:

In[6]:=6
✖
https://wolfram.com/xid/0btnz7gc2iv2-d3v70g
Out[6]=6

Experimentally, a CO molecule oscillates about its equilibrium length with an effective spring constant of k=TemplateBox[{1.86`, {"kN", , "/", , "m"}, kilonewtons per meter, {{(, "Kilonewtons", )}, /, {(, "Meters", )}}}, QuantityTF]. The oscillations will be governed by the quantum harmonic oscillator equation. In the following, is the reduced mass of the molecule, is the natural frequency, is the displacement from the equilibrium position, and is the reduced Planck's constant:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-551rrk

Compute the eigenvalues—the energies of the respective states—and normalized eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-jxv26o
Out[2]=2

If the particle is in an equal superposition of the four states, the wavefunction has the following form:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-qvnvu
Out[3]=3

Compute , , and using base units of atomic mass units, femtoseconds, and picometers, which give values close to order unity:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-oz2boj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-nbi4dx
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0btnz7gc2iv2-9qfa20
Out[6]=6

The response of the eigenfunctions to the potential energy can be visualized by rescaling them to fit in the band :

In[7]:=7
✖
https://wolfram.com/xid/0btnz7gc2iv2-rxoenq
Out[7]=7

The probability density function of the displacement from equilibrium is given by :

In[8]:=8
✖
https://wolfram.com/xid/0btnz7gc2iv2-mwva1k
Out[8]=8

As a probability distribution, the integral of over the reals is 1 for all :

In[9]:=9
✖
https://wolfram.com/xid/0btnz7gc2iv2-cqpldr
Out[9]=9

Visualize the probability density over time:

In[10]:=10
✖
https://wolfram.com/xid/0btnz7gc2iv2-s7tsiv
Out[10]=10

Properties & Relations  (6)Properties of the function, and connections to other functions

Use NDEigensystem to find numerical eigenvalues and eigenvectors:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-b2g6ab

Find exact eigenvalues and eigenvectors:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-h80wjy
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-bnu8sm
Out[3]=3

Find numerical eigenvalues and eigenvectors:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-fub3ug
Out[4]=4

Use DEigenvalues to find the eigenvalues for a differential operator:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-ls3dpn

Find eigenvalues and eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-efw01u
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-gdvfkv
Out[3]=3

Find eigenvalues only:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-cb55dh
Out[4]=4

Use DSolve to solve an eigenvalue problem:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-hoe9ag

Find eigenvalues and eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-bqrzjm
Out[2]=2

Find the complete eigensystem:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-es16mg
Out[3]=3

The eigenfunctions given by DEigensystem are orthogonal:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-bbit9o

Find eigenvalues and eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-i15m9d
Out[2]=2

Verify that the eigenfunctions are orthogonal:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-dfszts
Out[3]=3

The system of eigenfunctions given by DEigensystem is not orthonormal by default:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-gx3hrl

Find eigenvalues and eigenfunctions:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-uv309
Out[2]=2

The eigenfunctions are not orthonormalized by default:

In[3]:=3
✖
https://wolfram.com/xid/0btnz7gc2iv2-kwuivh

Use Method->"Normalize" to obtain an orthonormal system:

In[4]:=4
✖
https://wolfram.com/xid/0btnz7gc2iv2-gzxmv
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0btnz7gc2iv2-htdmtj

Apply N[DEigensystem[...]] to invoke NDEigensystem if symbolic evaluation fails:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-ktfv0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-logptw
Out[2]=2

Possible Issues  (2)Common pitfalls and unexpected behavior

Inhomogeneous Dirichlet conditions are replaced with homogeneous ones:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-d7hkcu
Out[1]=1

The same result:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-jg3nz5
Out[2]=2

Inhomogeneous Neumann values are replaced with homogeneous ones:

In[1]:=1
✖
https://wolfram.com/xid/0btnz7gc2iv2-vkj213
Out[1]=1

The same result:

In[2]:=2
✖
https://wolfram.com/xid/0btnz7gc2iv2-0cm5i9
Out[2]=2
Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.
✖
Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.

Text

Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.

✖
Wolfram Research (2015), DEigensystem, Wolfram Language function, https://reference.wolfram.com/language/ref/DEigensystem.html.

CMS

Wolfram Language. 2015. "DEigensystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DEigensystem.html.

✖
Wolfram Language. 2015. "DEigensystem." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DEigensystem.html.

APA

Wolfram Language. (2015). DEigensystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DEigensystem.html

✖
Wolfram Language. (2015). DEigensystem. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DEigensystem.html

BibTeX

@misc{reference.wolfram_2025_deigensystem, author="Wolfram Research", title="{DEigensystem}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/DEigensystem.html}", note=[Accessed: 22-June-2025 ]}

✖
@misc{reference.wolfram_2025_deigensystem, author="Wolfram Research", title="{DEigensystem}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/DEigensystem.html}", note=[Accessed: 22-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_deigensystem, organization={Wolfram Research}, title={DEigensystem}, year={2015}, url={https://reference.wolfram.com/language/ref/DEigensystem.html}, note=[Accessed: 22-June-2025 ]}

✖
@online{reference.wolfram_2025_deigensystem, organization={Wolfram Research}, title={DEigensystem}, year={2015}, url={https://reference.wolfram.com/language/ref/DEigensystem.html}, note=[Accessed: 22-June-2025 ]}