What is Eigenvalue problem: Definition and 85 Discussions

In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues


{\displaystyle \lambda }
, left eigenvectors


{\displaystyle y}
and right eigenvectors


{\displaystyle x}
such that





{\displaystyle Q(\lambda )x=0{\text{ and }}y^{\ast }Q(\lambda )=0,}












{\displaystyle Q(\lambda )=\lambda ^{2}A_{2}+\lambda A_{1}+A_{0}}
, with matrix coefficients











{\displaystyle A_{2},\,A_{1},A_{0}\in \mathbb {C} ^{n\times n}}
and we require that




{\displaystyle A_{2}\,\neq 0}
, (so that we have a nonzero leading coefficient). There are


{\displaystyle 2n}
eigenvalues that may be infinite or finite, and possibly zero. This is a special case of a nonlinear eigenproblem.


{\displaystyle Q(\lambda )}
is also known as a quadratic polynomial matrix.

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  1. G

    Put the eigenvalue function in self-adjoint form

    Here’s my work: The integrating factor I find is (x^(2)-1)^1/2. The self adjoint form I find is -d/dx (((1-x^(2))^(3/2))*dy/dx))=k(x^(2)-1)^(1/2). Am I right?
  2. CuriousLearner8

    A Eigenvalue Problem of Quantum Mechanics

    Hello, I hope you are doing well. I had a question about the eigenvalue problem of quantum mechanics. In a past class, I remember it was strongly emphasized that the eigenvalues of an eigenvalue problem is what we measure in the laboratory. ##A\psi = a\psi## where A would be the operator...
  3. A

    Engineering Book considering FEM analysis for complex eigenvalues (incl. damping)

    Can anyone recommend a book in which complex eigenvalue problems are treated? I mean the FEM analysis and the theory behind it. These are eigenvalue problems which include damping. I think that it is used for composite materials and/or airplane engineering (maybe wing fluttering?).
  4. Viona

    Operator acts on a ket and a bra using Dirac Notation

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  5. B

    I It seems that the eigenvalue problem rules out the possibility of E=0?

    Since the eigenvalue problem can't distinguish between a non-existent wavefunction (and therefore a non-existent particle), and the energy being zero. This is the next thing that has started bothering me on my journey to understand quantum mechanics. For example, in the algebraic derivation of...
  6. Andrew1235

    Finding the directions of eigenvectors symmetric eigenvalue problem

    In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
  7. P

    A Eigenvalues of block matrix/Related non-linear eigenvalue problem

    I have a matrix M which in block form is defined as follows: \begin{pmatrix} A (\equiv I + 3\alpha J) & B (\equiv -\alpha J) \\ I & 0 \end{pmatrix} where J is an n-by-n complex matrix, I is the identity and \alpha \in (0,1] is a parameter. The problem is to determine whether the eigenvalues of...
  8. T

    I Simple Generalized Eigenvalue problem

    Good Morning Could someone give me some numbers for a Generalized EigenValue problem? I have lots of examples for a 2 x 2, but would like to teach the solution for a 3x3. I would prefer NOT to turn to a computer to solve for the characteristic equation, but would like an equation where the...
  9. M

    A Eigenvalue problem: locating complex eigenvalues via frequency scan

    Hi PF! Here's an ODE (for now let's not worry about the solutions, as A LOT of preceding work went into reducing the PDEs and BCs to this BVP): $$\lambda^2\phi-0.1 i\lambda\phi''-\phi'''=0$$ which admits analytic eigenvalues $$\lambda =-2.47433 + 0.17337 I, 2.47433 + 0.17337 I, -10.5087 +...
  10. T

    I Eigenvalue Problem -- Justification

    Hello! Suppose you have two masses, that are connected by a spring. Each mass is, in turn, connected by a spring to a wall So there is a straight line: left wall to first mass, first mass to second mass, second mass to right wall This problem can be analyzed as an eigenvalue problem. We...
  11. bluesky314

    I Question about an eigenvalue problem: range space

    A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular. In the proof, he defines U=range(T-𝜆I) (as we have...
  12. peguerosdc

    I Confusion with Dirac notation in the eigenvalue problem

    Hi! I am studying Shankar's "Principles of QM" and the first chapter is all about linear algebra with Dirac's notation and I have reached the section "The Characteristic Equation and the Solution to the Eigenvalue Problem" which says that starting from the eigenvalue problem and equation 1.8.3...
  13. M

    Quadratic eigenvalue problem and solution (solved in Mathematica)

    Hi PF! Given the quadratic eigenvalue problem ##Q(\lambda) \equiv (\lambda^2 M + \lambda D + K)\vec x = \vec 0## where ##K,D,M## are ##n\times n## matrices, ##\vec x## a ##1\times n## vector, the eigenvalues ##\lambda## must solve ##\det Q(\lambda)=0##. When computing this, I employ a...
  14. M

    MATLAB Solving Polynomial Eigenvalue Problem

    Hi PF! I'm trying to solve the polynomial eigenvalue problem ##M \lambda^2 + \Phi \lambda + K## such that K = [5.92 -.9837;-0.3381 109.94]; I*[14.3 24.04;24.04 40.4]; M = [1 0;0 1]; [f lambda cond] = polyeig(M,Phi,K) I verify the output of the first eigenvalue via (M*lambda(1)^2 +...
  15. S

    Eigenvalue problem -- Elastic deformation of a membrane

    Homework Statement An elastic membrane in the x1x2-plane with boundary circle x1^2 + x2^2 = 1 is stretched so that point P(x1,x2) goes over into point Q(y1,y2) such that y = Ax with A = 3/2* [2 1 ; 1 2] find the principal directions and the corresponding factors of extension or contraction of...
  16. M

    Mathematica Eigenvalue problem and badly conditioned matrices

    Hi PF! I am trying to solve the eigenvalue problem ##A v = \lambda B v##. I thought I'd solve this by $$A v = \lambda B v \implies\\ B^{-1} A v = \lambda v\implies\\ (B^{-1} A - \lambda I) v = 0 $$ and then using the built in function Eigenvalues and Eigenvectors on the matrix ##B^{-1}A##. But...
  17. M

    A Solving an ODE Eigenvalue Problem via the Ritz method

    Hi PF! I want to solve ##u''(x) = -\lambda u(x) : u(0)=u(1)=0##. I know solutions are ##u(x) = \sin(\sqrt{\lambda} x):\lambda = (n\pi)^2##. I'm trying to solve via the Ritz method. Here's what I have: define ##A(u)\equiv d^2_x u## and ##B(u)\equiv u##. Then in operator form we have ##A(u) =...
  18. M

    A Eigenvalue Problem and the Calculus of Variations

    Hi PF! Given ##B u = \lambda A u## where ##A,B## are linear operators (matrices) and ##u## a function (vector) to be operated on with eigenvalue ##\lambda##, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of ##(Bu,u)## subject to ##(Au,u)=1##...
  19. A

    MHB A question on matrix's eigenvalue problem from Eberhard Zeidler's first volume of Nonlinear Function

    The question is posted in the following post in MSE, I'll copy it here: https://math.stackexchange.com/questions/1407780/a-question-on-matrixs-eigenvalue-problem-from-eberhard-zeidlers-first-volume-o I have a question from Eberhard Zeidler's book on Non-Linear Functional Analysis, question...
  20. C

    Sturm-Liouville Eigenvalue Problem (Variational Method?)

    Homework Statement Using Sturm-Liouville theory, estimate the lowest eigenvalue ##\lambda_0## of... \frac{d^2y}{dx^2}+\lambda xy = 0 With the boundary conditions, ##y(0)=y(\pi)=0## And explain why your estimate but be strictly greater than ##\lambda_0##Homework Equations ##\frac{d}{dx} \left...
  21. Mr Davis 97

    Eigenvalue Problem: Show 0 is the Only Eigenvalue of A When A^2=0

    Homework Statement Let ##A## be an ##n \times n## matrix. Show that if ##A^2## is the zero matrix, then the only eigenvalue of ##A## is 0. Homework EquationsThe Attempt at a Solution All eigenvalues and eigenvectors must satisfy the equation ##A\vec{v} = \lambda \vec{v}##. Multiplying both...
  22. Hamza Abbasi

    I Eigenvalue Problem: What Is It?

    While reading problems in my physics book , I encountered a statement very often "Eigen Value Problem" , I read about it from many sources , but wasn't able to understand it . So what exactly is an Eigen Value Problem?
  23. I

    Eigenvalue Problem: Find All Eigen-Values & Eigen-Fns

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  24. J

    A Large scale eigenvalue problem solver

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  25. M

    How to apply boundary condition in generalized eigenvalue problem?

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  26. M

    MHB Eigenvalue problem of the form Sturm-Liouville

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  27. K

    Eigenvalue problem with nonlocal condition

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  28. maajdl

    Singular value decomposition and eigenvalue problem:

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  29. N

    Understanding the Eigenvalue Problem for a 4x4 Matrix with Rank 1 and Trace 10

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  30. H

    Quick question about solving an eigenvalue problem

    I just have a question about the problem for when the eigenvalue = 0 Homework Statement for y_{xx}=-\lambda y with BC y(0)=0 , y'(0)=y'(1) Homework Equations The Attempt at a Solution y for lamda = 0 is ax+b so from BC: y(0)=b=0 and a=a What is the conclusion to...
  31. T

    What is an eigenvalue problem?

    Are eigenvalue problems and boundary value problems (ODEs) the same thing? What are the differences, if any? It seems to me that every boundary value problem is an eigenvalue problem... Is this not the case?
  32. V

    Two-Degree-Of-Freedom Linear System: Eigenvalue problem

    I've found the characteristic equation of the system I'm trying to solve: $$ω^{4}m_{1}m_{2}-k(m_{1}+2m_{2})ω^{2}+k^{2}=0$$ I now need to find the eigenfrequencies, i.e. the two positive roots of this equation, and then find the corresponding eigenvectors. I've been OK with other examples...
  33. P

    Reducing angular Schrodinger equation to eigenvalue problem

    Homework Statement The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment {\bf d}=d{\bf \hat{k}} is, in spherical polar coordinates (r,\vartheta,\varphi), \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial...
  34. J

    Eigenvalue problem and initial-value problem?

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  35. W

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    Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem: \begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu +...
  36. P

    Solving Eigenvalue Problem After Galerkin

    Eigenvalue problem after galerkin Homework Statement i am working on vibration of cylindrical shell analysis, after solving the equations of motion by galerkin method , i reach to an eigenvalue problem this way : { p^2*C1+p*C2+C3 } * X=0 C1,C2,C3 are all square matrices of order n*n...
  37. J

    How Do You Solve Modified Eigenvalue Problems Like Lq=\lambda q + a?

    I know that eigenvalue problem like Lq=\lambda q could be easily solved by eig command in Matlab. But how to solve a problem like Lq=\lambda q + a, where a has the same dimension with the eigenfunction q? Thanks a lot in advance. Jo
  38. D

    Sturm-Liouville Eigenvalue problem

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  39. Y

    Help finding a general solution for an eigenvalue problem

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  40. C

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  41. J

    Generalized Eigenvalue Problem

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  42. R

    Verification sequence of eigenvalue problem

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  43. H

    Solving Eigenvalue Problem with Periodic BCs: Find b for Self-Adjointness

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  44. D

    Coupled non-homogenous eigenvalue problem help?

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  45. M

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  46. 3

    Do Rotational Matrices Always Yield Real Eigenvalues?

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  47. C

    What is the eigenvalue problem for the given matrix and how can it be solved?

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  48. Hootenanny

    Mathematica Generalised Eigenvalue Problem in Mathematica

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  49. M

    Multilinear eigenvalue problem

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  50. A

    True or false eigenvalue problem

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