Eigenvalues Definition and 853 Threads

In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by



λ


{\displaystyle \lambda }
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.

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  1. R

    I The Orthogonality of the Eigenvectors of a 2x2 Hermitian Matrix

    The eigenvectors of a hermitian matrix corresponding to unique eigenvalues are orthogonal. This is not too difficult of a statement to prove using mathematical induction. However, this case is seriously bothering me. Why is the dot product of the vectors not rightly zero? Is there something more...
  2. T

    I Unravelling Structure of a Symmetric Matrix

    Hey guys, I was wondering if anyone had any thoughts on the following symmetric matrix: $$\begin{pmatrix} 0.6 & 0.2 & -0.2 & -0.6 & -1\\ 0.2 & -0.2 & -0.2 & 0.2 & 1\\ -0.2 & -0.2 & 0.2 & 0.2 & -1\\ -0.6 & 0.2 & 0.2 & -0.6 & 1\\ -1 & 1 & -1 & 1 & -1 \end{pmatrix} $$ Notably, when one solves for...
  3. T

    I Generalized Eigenvalues of Pauli Matrices

    Consider a generic Hermitian 2x2 matrix ##H = aI+b\sigma_{x}+c\sigma_{y}+d\sigma_{z}## where ##a,b,c,d## are real numbers, ##I## is the identity matrix and ##\sigma_{i}## are the 2x2 Pauli Matrices. We know that the eigenvalues for ##H## is ##d\pm\sqrt{a^2+b^2+c^2}## but now suppose I have the...
  4. cianfa72

    I Eigenstates of particle with 1/2 spin (qbit)

    A very basic doubt about a QM system (particle) with spin 1/2 (qbit). From the Bloch sphere representation we know that a qbit's pure state is represented by a point on the surface of the sphere. Picking a base, for instance the pair of vector/states ##\ket{\uparrow}## and ##\ket{\downarrow}##...
  5. J

    Eigenvalues of Hamiltonian operator

    Hello, I try to solve this problem, and I think a) wasn't too hard, I have the following solution: ##H = \lambda (\frac{\vec{S^2-(\vec{S_1}^2+\vec{S_2}^2)}{2})##. I struggle with 2. I find it very abstract. When I have H as a matrix I know how to calculate eigenvalues, but I don't know how...
  6. D

    I Commutators, operators and eigenvalues

    Hi I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is...
  7. K

    A Parameter optimization for the eignevalues of a matrix

    Hello! I have a matrix (about 20 x 20), which corresponds to a given Hamiltonian. I would like to write an optimization code that matches the eigenvalues of this matrix to some experimentally measured energies. I wanted to use gradient descent, but that seems to not work in a straightforward...
  8. T

    I How can I test for positive semi-definiteness in matrices?

    On a side note I'm posting on PF more frequently as I have exams coming and I need some help to understand some concepts. After my exams I will probably go inactive for a while. So I'll get to the point. Suppose we have a matrix A and I wish to check if it is positive semi definite. So one easy...
  9. A

    Engineering How do I find the transition matrix of this dynamic system?

    Hello! I have the following matrix (picture 1.)and I am susposed to find the transition matrix ($$ \phi $$) now for that I need the eigenvalue and vectors of this matrix A. The eigenvalues are 1,1 and 2. The eigenvectors I have found to be (1 0 0) (1 1 0) (5 3 1). Now to find the transition...
  10. L

    Model for a Qubit system using the Hamiltonion Operator

    Hi, unfortunately, I am not sure if I have calculated the task a correctly. I calculated the eigenvalues with the usual formula ##\vec{0}=(H-\lambda I) \psi## and got the following results $$\lambda_1=E_1=-\sqrt{B^2+\nabla^2}$$ $$\lambda_2=E_2=\sqrt{B^2+\nabla^2}$$ I'm just not sure about...
  11. ergospherical

    Degenerate Perturbation: Calculating Eigenvalues

    Say a model hamiltonian with unperturbed eigenvalues E1 and E2 = E3 is subjected to a perturbation V such that V12 = V21 = x and V13 = V31 = x2, with all other elements zero. I'm having trouble calculating the corrected eigenvalues. In the degenerate subspace spanned by |2> and |3> I need to...
  12. M

    Finding eigenvalues and eigenvectors given sub-matrices

    For this, The solution is, However, does someone please know what allows them to express the eigenvector for each of the sub-matrix in terms of t? Many thanks!
  13. M

    I Find the Eigenvalues and eigenvectors of 3x3 matrix

    Assume a table A(3x3) with the following: A [ 1 2 1 ]^T = 6 [ 1 2 1 ]^T A [ 1 -1 1 ]^T = 3 [ 1 -1 1 ]^T A [ 2 -1 0]^T = 3 [ 1 -1 1]^T Find the Eigenvalues and eigenvectors: I have in mind to start with the Av=λv or det(A-λI)v=0.... Also, the first 2 equations seems to have the form Av=λv...
  14. M

    Proving eigenvalues of a 2 x 2 square matrix

    For this, Does someone please know why the equation highlighted not be true if ##(A - 2I_2)## dose not have an inverse? Many thanks!
  15. M

    Using inverse to find eigenvalues

    For this, I don't understand how if ##(A - 2I_2)^{-1}## has an inverse then the next line is true. Many thanks!
  16. L

    Mathematica How do you calculate determinants and eigenvalues in Mathematica?

    Hi, In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment. Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...
  17. M

    Diagonalizing a Matrix: Understanding the Process and Power of Matrices

    For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!
  18. S

    Condition such that the symmetric matrix has only positive eigenvalues

    My attempt: $$ \begin{vmatrix} 1-\lambda & b\\ b & a-\lambda \end{vmatrix} =0$$ $$(1-\lambda)(a-\lambda)-b^2=0$$ $$a-\lambda-a\lambda+\lambda^2-b^2=0$$ $$\lambda^2+(-1-a)\lambda +a-b^2=0$$ The value of ##\lambda## will be positive if D < 0, so $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...
  19. H

    If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

    Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...
  20. K

    I Velocity operator, its expression and eigenvalues

    Cohen Tannoudji pp 215 pp 225 pp 223 From above we can say that there exists a velocity operator ##\mathbf v=\frac{\mathbf p}{m}## ,whose eigenvalues are the observed values of velocity. 1. I've seen multiple times that we can't define velocity in quantum mechanics, but here I find that...
  21. codebpr

    A Does the Maximum Lyapunov exponent depend on the eigenvalues?

    I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
  22. T

    I Getting eigenvalues of an arbitrary matrix with programming

    I have learnt about the power iteration for any matrix say A. How it works is that we start with a random compatible vector v0. We define vn+1 as vn+1=( Avn)/|max(Avn)| After an arbitrary large number of iterations vn will slowly converge to the eigenvector associated with the dominant...
  23. hilbert2

    A Changing Hamiltonian with some eigenvalues constant

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  24. T

    MATLAB FEM, Matlab and the modes of an element

    Hello I wrote a Matlab code to form the 8 by 8 stiffness matrix of a single, 4-noded element, for a plane strain problem for an isotropic element. I conduct an eigenvalue analysis on this matrix Matlabe reports 5 non-zero eigenvalue modes, and 3 zero-eigenvalue modes (as expected) Of the 3...
  25. P

    X^4 perturbative energy eigenvalues for harmonic oscillator

    The book(Schaum) says the above is the solution but after two hours of tedious checking and rechecking I get 2n^2 in place or the 3n^2. Am I missing something or is this just a typo?
  26. ergospherical

    I Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

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  27. Salmone

    I Solving a Particle on the Surface of a Sphere: Obtaining Eigenvalues

    The Hamiltonian of a particle of mass ##m## on the surface of a sphere of radius ##R## is ##H=\frac{L^2}{2mR^2}## where ##L## is the angular momentum operator. I want to solve the TISE ##\hat{H}\psi=E\psi## and in order to do that I rewrite ##L^2## in Schroedinger's representation in spherical...
  28. H

    Prove that ##\lambda## or ##-\lambda## is an eigenvalue for ##T##.

    The statement " If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2##", means that there exists an ##x## in ##V## such that ## T^2 (x) = \lambda^2 x##. If ##T(x) = \mu x##, we've have $$ T [T(x)]= T ( \mu x)$$ $$ T^2 (x) = \mu^2 x$$ $$ \lambda ^2 = \mu ^2...
  29. nomadreid

    I This is an invalid argument about eigenvalues, but why?

    The fallacy in the summary is not covered in the sites discussing eigenvalues, so there must be something blindingly and embarrassingly obvious that is wrong. I would be grateful if someone would point it out. Thanks.
  30. S

    I Multiple questions about eigenstates and eigenvalues

    I have multiple questions about eigenstates and eigenvalues. The Hilbert space is spanned by independent bases. The textbook said that the eigenvectors of observable spans the Hilbert space. Here comes the question. Do the eigenvectors of multiple observables span the same Hilber space? Here...
  31. J

    Discretizing a 1D quantum harmonic oscillator, finding eigenvalues

    ##x## can be discretized as ##x \rightarrow x_k ## such that ##x_{k + 1} = x_k + dx## with a positive integer ##k##. Throughout we may assume that ##dx## is finite, albeit tiny. By applying the Taylor expansion of the wavefunction ##\psi_n(x_{k+1})## and ##\psi_n(x_{k-1})##, we can quickly...
  32. 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?).
  33. N

    I Find the center manifold of a 2D system with double zero eigenvalues

    I have to find the center manifold of the following system \begin{align} \dot{x}_1&=x_2 \\ \dot{x}_2&=-\frac{1}{2}x_1^2 \end{align} which has a critical point at ##x_0=\begin{bmatrix}0 & 0\end{bmatrix}##. Its linearization at that point is \begin{align} D\mathbf {f}(\mathbf {x_0}) =...
  34. R

    Why Are My Coupled Oscillator Eigenvalues Incorrect?

    Hi, I have to find the eigenvalues and eigenvectors for a system of 3 masses and 4 springs. At the end I don't get the right eigenvalues, but honestly I don't know why. Everything seems fine for me. I spent the day to look where is my error, but I really don't know. ##m_a = m_b = m_c## I got...
  35. L

    Prove eigenvalues of the derivatives of Legendre polynomials >= 0

    The problem has a hint about finding a relationship between ##\int_{-1}^1 (P^{(k+1)}(x))^2 f(x) dx## and ##\int_{-1}^1 (P^{(k)}(x))^2 g(x) dx## for suitable ##f, g##. It looks they're the weighting functions in the Sturm-Liouville theory and we may be able to make use of Parseval's identity...
  36. BWV

    Negative eigenvalues in covariance matrix

    Trying to run the factoran function in MATLAB on a large matrix of daily stock returns. The function requires the data to have a positive definite covariance matrix, but this data has many very small negative eigenvalues (< 10^-17), which I understand to be a floating point issue as 'real'...
  37. Wannabe Physicist

    Find the eigenfunction and eigenvalues of ##\sin\frac{d}{d\phi}##

    Here is what I tried. Suppose ##f(\phi)## and ##\lambda## is the eigenfunction and eigenvalue of the given operator. That is, $$\sin\frac{d f}{d\phi} = \lambda f$$ Differentiating once, $$f'' \cos f' = \lambda f' = f'' \sqrt{1-\sin^2f'}$$ $$f''\sqrt{1-\lambda^2 f^2} = \lambda f'$$ I have no...
  38. F

    Problem calculating eigenvalues and eigenvectors

    Hello everyone. I am trying to construct a functioning version of randomfields (specifically 2D_karhunen_loeve_identification_example.py) in Matlab. For that, I have to calculate the Karhunen-Loève expansion of 2D data, since this is what it says in the documentation. I also have some sample...
  39. Haorong Wu

    I Improper density matrix with negative eigenvalues

    Hi, there. I am working with a model, in which the dimension of the Hilbert space is infinite. But Since only several states are directly coupled to the initial state and the coupling strength are weak, then I only consider a subspace spanned by these states. The calculation shows that the...
  40. S

    Diagonalizing a matrix given the eigenvalues

    The following matrix is given. Since the diagonal matrix can be written as C= PDP^-1, I need to determine P, D, and P^-1. The answer sheet reads that the diagonal matrix D is as follows: I understand that a diagonal matrix contains the eigenvalues in its diagonal orientation and that there must...
  41. StenEdeback

    Why Are There Additional Eigenvalues and Eigenvectors of J3?

    The J3 matrix of two dimensional SU2 consists of two row vectors (1 0) and (0 -1). When I calculate the eigenvalues of an eigenvector v the usual way with J3v=kv then I find eigenvalues +-1 and eigenvectors (1 0) and (0 1). But how is it possible to say that there are other eigenvectors and...
  42. W

    MHB Find Eigenvalues & Basis C2 Matrix: Help!

    Good afternoon to all again! I'm solving last year's problems and can't cope with this problem:( help me to understand the problem and find a solution!
  43. lelouch_v1

    A Eigenvalues of Hyperfine Hamiltonian

    I was reading a paper on Radical-Pair mechanism (2 atoms with 1 valence electron each) and the author used the hyperfine hamiltonian $$H_{B}=-B(s_{D_z}+s_{A_z})+As_{D_x}I_x+As_{D_y}I_y+as_{D_z}I_z$$ and found the following eigenvalues: a/4 (doubly degenerate) , a/4±B , (-a-2B±2√(A^2+B^2)) ...
  44. H

    Find the eigenvalues of a 3x3 matrix

    Hi, I have a 3 mass system. ##M \neq m## I found the forces and I get the following matrix. I have to find ##\omega_1 , \omega_2, \omega_3## I know I have to find the values of ##\omega## where det(A) = 0, but with a 3x3 matrix it is a nightmare. I can't find the values. I'm wondering if...
  45. 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...
  46. T

    A What is the purpose of modal analysis?

    Hello all, I have been asking this question, here, and gaining more insight. I think I can finally ask it the way I need. I can: Conduct an eigenvalue analysis Code the Lanczos algorithm. Understand mode shapes Build the solution of set of coupled differential equations from mode shapes...
  47. K

    What can we say about the eigenvalues if ##L^2=I##?

    This was a problem that came up in my linear algebra course so I assume the operation L is linear. Or maybe that could be derived from given information. I don't know how though. I don't quite understand how L could be represented by anything except a scalar multiplication if L...
  48. The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses

    The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses

    Zach Star gives an explanation of Eigenvalues and Eigenvectors, with some applications
    • scottdave
    • Media item
    • applied mathematics eigenvalues eigenvectors math and physics
    • Comments: 0
    • Category: Linear Algebra
  49. L

    A What Does the Book Say About the Eigenvalues of 3x3 Matrices?

    I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.
  50. 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 +...
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