What is Eigenvalues: Definition and 851 Discussions

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. Dusty912

    Finding eigenvectors with eigenvalues

    Homework Statement So just curious about a specific problem that I am worries about running into on my test tomorrow. When trying to find eigen vectors with the eigen values what is there is a discrepancy between the two systems obtained after doing the matrix arithmetic? such as after using...
  2. P

    MHB Effie's question via email about Eigenvalues, Eigenvectors and Diagonalisation

    Effie has correctly found that the eigenvalues of $\displaystyle \begin{align*} A = \left[ \begin{matrix} \phantom{-}3 & \phantom{-}2 \\ -3 & -4 \end{matrix} \right] \end{align*}$ are $\displaystyle \begin{align*} \lambda_1 = -3 \end{align*}$ and $\displaystyle \begin{align*} \lambda_2 = 2...
  3. P

    3D Harmonic Oscillator - Eigenfunctions and Eigenvalues

    Homework Statement Due to the radial symmetry of the Hamiltonian, H=-(ħ2/2m)∇2+k(x^2+y^2+z^2)/2 it should be possible to express stationary solutions to schrodinger's wave equation as eigenfunctions of the angular momentum operators L2 and Lz, where...
  4. M

    Confusion about eigenvalues of an operator

    Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This...
  5. A

    Energy eigenvalue and mass inverse relation?

    So, after time-independent 1D Schrodinger equation is solved, this is obtained E = n2π2ħ2/(2mL2) This means that the mass of the 'particle' is inversely related to the energy eigenvalue. Does this mean that the actual energy of the particle is inversely related to its mass? Isn't this counter...
  6. TheMathNoob

    Relation between complex eigenvalues and rotations

    Homework Statement I have the following matrix: 0 0 0 1 1 0 0 0 = A 0 1 0 0 0 0 1 0 and the vector v = (1,0,0,0) If I perform Av, this gives: Av=(0,1,0,0) And If I keep multiplying the result by A like A*A*(Av), the outcome will be something like j= (0,0,1,0) k=(0,0,0,1) l=(1,0,0,0) The...
  7. entropy1

    QM interpretations defined as eigenvalues?

    And back again with a strange/odd layman question: Actually, this remark of Jilang is a perfect illustration of what I am wondering about: It seems as if realism and locality are behaving like two eigenstates in a space of interpretations of QM. Most people seem to suggest that you must have...
  8. Mark44

    Insights What Are Eigenvectors and Eigenvalues? - Comments

    Mark44 submitted a new PF Insights post What Are Eigenvectors and Eigenvalues? Continue reading the Original PF Insights Post.
  9. j3dwards

    Why is the product of eigenvalues equal to the det(A)?

    Homework Statement Explain in your own words why the product of eigenvalues of any diagonalisable N × N matrix A must equal the determinant of A. Homework Equations MT=M-1 The Attempt at a Solution So what I do know: the determinant measures the change in area of the unit square under the...
  10. Y

    MHB Finding Eigenvalues of Matrix A: Wrong Answer, What Am I Doing Wrong?

    Hello all, I have a matrix A and I am looking for it's eigenvalues. No matter what I do, I find that the eigenvalues are 0, 1 and (k+1), while the answer of both the book and Maple is 0 and (k+2). I tried two different technical approaches, both led to the same place. The matrix is...
  11. F

    Find the Eigenvalues and Eigenvectors of 4x4 Matrix.

    Homework Statement X= 1st row: (0, 1, 0, 0), 2nd row: (1, 0, 0, 0), 3rd row: (0, 0, 0, 1-i), 4th row: (0, 0, 1+i, 0) Find the eigenvalues and eigenvectors of the matrix X. Homework Equations |X-λI|=0 (characteristic equation) (λ is the eigenvalues, I is the identity matrix) (X-λI)V=0 (V is the...
  12. S

    How Do You Find Eigenvalues and Eigenvectors for a Linear Transformation?

    Homework Statement Given the linear transformation l : R 2 → R 2 defined below, find characteristic equation, real eigenvalues and corresponding eigenvectors. a) l(x, y) = (x + 5y, 2x + 4y) Homework Equations characteristic equation = det (A-λI) = 0 The Attempt at a Solution l(x, y) = (x +...
  13. P

    Perturbed Hamiltonian and its affect on the eigenvalues

    Homework Statement Homework Equations $$E_n^{(2)}=\sum_{k\neq n}\frac{|H_{kn}'|^2}{E_n^{(0)}-e_k^{(0)}}$$ The Attempt at a Solution Not sure where to start here. The question doesn't give any information about the unperturbed Hamiltonian. Some guidance on the direction would be great...
  14. Y

    MHB Why Are Eigenvalues and Eigenvectors Important in Linear Algebra?

    Hello all I have a theoretical question. I know how to find the eigenvalues and eigenvectors of a matrix A. What I am not sure about, is what it all means and why do we need it for. I did some reading, and saw something about stretching vector, if I not mistaken, if I have a vector v, and I...
  15. bcrowell

    Eigenvalues of curvature tensors as curvature scalars?

    I've been playing around with the Carminati-McLenaghan invariants https://en.wikipedia.org/wiki/Carminati–McLenaghan_invariants , which are a set of curvature scalars based on the Riemann tensor (not depending on its derivatives). In general, we want curvature scalars to be scalars that are...
  16. RJLiberator

    Non-degenerate Hermitian Matrices and their Eigenvalues

    Homework Statement Is there a non-degenerate 2x2 matrix that has only real eigenvalues but is not Hermitian? (Either find such a matrix, or prove that it doesn't exist) Homework EquationsThe Attempt at a Solution Here's my problem. I'm getting Contradicting results. So, I found this 2x2...
  17. N

    Distribution of 2 matrices with the same eigenvalues

    Hi, I was wondering if two matrices with the same eigenvalues share the same PDF. Any ideas and/or references would be helpful. Thanks in advance
  18. ognik

    MHB Show that the eigenvalues of any matrix are unaltered by a similarity transform

    Show that the eigenvalues of any matrix are unaltered by a similarity transform - the book says this follows from the invariance of the secular equation under a similarity transform - which is news to me. The secular eqtn is found by Det(A-\lambda I)=0 and is a poly in \lambda , so I can't see...
  19. kostoglotov

    Why are the eigenvectors the axes of an ellipse?

    I'm almost there in terms of understanding it, but I need to go beyond the text. Here is the example problem: imgur link: http://i.imgur.com/UMj55tF.jpg I can see that where we have 1 = \vec{x}^T A \vec{x} = \lambda \vec{x}^T \vec{x} that 1=\lambda \vec{x}^T \vec{x} = \lambda ||\vec{x}||^2...
  20. W

    Eigenvalues of a 2x2 Matrix: What's the Mistake?

    Homework Statement Find the eigenvalues of the matrix ## \left( \begin{array}{cc} 3 & -1.5\\ -1.5 & -1\\ \end{array} \right) ## It's probably a really stupid mistake, but the answer I get doesn't match the answer from wolfram alpha's eigenvalue calculator... always a bad sign. Homework...
  21. Y

    Proving eigenvalues and diagonalizability

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > Let A be a square matrix of order n such that A^2 = I a) Prove that if -1 is the only eigenvalue of A, then A= -I b) Prove that if 1 is the only eigenvalue of A, then A= I c) Prove that A is...
  22. kostoglotov

    How can e^{Diag Matrix} not be an infinite series?

    So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...
  23. kostoglotov

    Matrix with repeated eigenvalues is diagonalizable....?

    MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable". imgur: http://i.imgur.com/Q4pbi33.jpg and imgur: http://i.imgur.com/RSOmS2o.jpg Upon rereading...I do see the possibility...
  24. S

    MHB Software for calculating eigenvalues and eigenfunctions of an integral operator

    Hi can someone direct me to a free software to calculate eigenvalues and normalized eigenfunctions of a linear integral operator. I am trying to solve a fredholm integral equation with degenerate kernel using it instead of linear equations thanks sarrah
  25. I

    Self adjoint operators, eigenfunctions & eigenvalues

    Homework Statement Consider the space ##P_n = \text{Span}\{ e^{ik\theta};k=0,\pm 1, \dots , \pm n\}##, with the hermitian ##L^2##-inner product ##\langle f,g\rangle = \int_{-\pi}^\pi f(\theta) \overline{g(\theta)}d\theta##. Define operators ##A,B,C,D## as ##A = \frac{d}{d\theta}, \; \; B=...
  26. D

    Find eigenvalues and eigenvectors of weird matrix

    Homework Statement find eigenvalues and eigenvectors for the following matrix |a 1 0| |1 a 1| |0 1 a| Homework EquationsThe Attempt at a Solution I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3 - a...
  27. AwesomeTrains

    Eigenvalues of disturbed Hamiltonian

    Hello everyone! I'm trying to follow a solution to a problem from the book "Problems and Solutions on Quantum Mechanics", it's problem 1017. There's a step where they go on too fast, and I can't follow. I've posted the solution and where my problem is down below. Homework Statement The dynamics...
  28. Diffie Heltrix

    Norm indueced by a matrix with eigenvalues bigger than 1

    Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...
  29. H

    What is the relationship between eigenvalues and eigenvectors in 3x3 matrices?

    What does it mean when it says eigenvalues of Matrix (3x3) A are the square roots of the eigenvalues of Matrix (3x3) B and the eigenvectors are the same for A and B?
  30. S

    Eigenvalues are invariant but eigenvectors are not

    Hi there. How would I show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors. Show also that the eigenvectors of a matrix are not an invariant. Explain why the dependence of the...
  31. A

    Finding rotation axis and angle, using eigenvalues

    http://im54.gulfup.com/ZtJZd5.png http://im54.gulfup.com/SJgJXh.png Till now compared with other subjects that I studied by my self, linear algebra is really the toughest one :bow: Anyway, here I found a plane using characteristic eq. which really surprised me since det(M)=1, M is the matrix...
  32. Tzabcan

    Eigenvalue and vector quick question

    So, I have the matrix: A = -1 -3 3 9 Eigenvalues i calculated to be λ = 8 and 0 Now when i calculate the Eigenvector for λ = 8, i get the answer -1 3 Then when solve for...
  33. Thor90

    Finding eigenvalues with QR method

    Hi, I am trying to solve the problem of finding eigenvalus for a general square symmetric matrix with the QR algorithm. I have understood that this task is much easier if the matrix is in an Hessemberg form, so I have implemented a function that does that with the Housholder method, but I can't...
  34. K

    Physical significance of Eigenvalues and Eigenvector?

    I want to know what exactly Eigen value imply. What is its Physical significance ? Physical significance of eigen vector? Does eigen value concept apply in signal processing or evalvating frequency response off a system?
  35. ATY

    Find the Spin Eigenvalues for Two Particles with Spin=1

    Hey guys, I really need your help. I have an exam tomorrow and no idea how to solve this task. Sorry for my bad english (did not use it for a long time). I hope that I translated the task correctly so that you might be able to help me. So We have two particles with spin=1 which means they have...
  36. J

    Find a 2x2 Matrix with given EigenValues

    Homework Statement Find a 2X2 matrix that has all non-zero entries where 3 is an eigenvalue Homework EquationsThe Attempt at a Solution well since the 2x2 matrix cannot be triangular, it makes things harder for me. I have no idea where to start. I am not given any eigenvectors either. It seems...
  37. A

    Understanding: Eigenvalues & Eigenvectors/Diagonalizing

    Hello, I'm having problem understanding this particular part, don't know it seems too dry and behind my capabilities of imagining the problems!, in the same time I feel like there is too many gaps in the way that the book explain the subject. I'm using "Mathematical methods in the physical...
  38. L

    Having difficulty working out a Char. Pol. for Eigenvalues

    Given the following matrix: 2 3 0 3 2 4 0 4 2 I'm having a difficult time working out the characteristic polynomial. I used the shortcut that I saw on YT, where it is (I am using x instead of lambda) X^3 - (trace) X^2 + (A11+A22+A33) X - DET(A) I got the following: trace is just 2+2+2 = 6...
  39. M

    Eigenvalues and eigenvectors, pauli matrices

    Homework Statement Look at the matrix: A = sin t sin p s_x + sin t sin p s_y +cos t s_z where s_i are the pauli matrices a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)? b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the...
  40. Angelos K

    LAPACK dgeev: parameter had illegal value

    Mod note: I revised the code below slightly, changing the loop control variable i to either j or k. The reason for this is that the browser mistakes the letter i in brackets for the BBCode italics tag, which causes some array expressions to partially disappear. Hello, I am trying for the first...
  41. E

    Exploring Quantum Eigenvalues: Algorithms for Extracting and Storing Information

    Quantum Eigenvalues have more information contents than non-quantum eigenvalues. What are the different algorithm to extract information from them? For example. If memory chips were made of quantum eigenvalues. How do you read and store information into them?
  42. binbagsss

    Verify eigenvalues of a TST matrix

    Homework Statement I have ##A=TST(-1,2-1),## and I need to show that an eigenvector of A is,##Y_{j}=sin(kj \pi / J).## and then find the full set of eigenvalues of A. The matrix A comes from writing ##-U_{j-1}+2U-U_{j+1}=h^{2}f(x_{j}), 1\le j \le J-1##, in the form ##AU=b## Homework...
  43. Nono713

    MHB Almost all matrices have n distinct eigenvalues

    Consider the following problem: I would like to solve this problem using only elementary techniques, along with the fact that almost all $n \times n$ complex matrices are invertible (obviously we don't use the fact that almost all matrices are diagonizable, ). Here is my approach: Let $A$ be...
  44. O

    Problem With solving the Eigenvalues of Tight Binding Method

    Dear All, this is my first post on this forum and hopefully I can get what I want. . I am trying to build the band structure of graphene using the tight-binding method based on slater-koster correction. I use a special code to construct both Hamiltonian and overlapping matrix. I have seen when...
  45. J

    4x4 Matrix Eigenvalues and Eigenvectors

    Homework Statement I have 4 equations. 3x+6y-5z-t=-8 6x-2y+3z+2t=13 4x-3y-z-3t=-1 5x+6y-3z+4t=-6 I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2 Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors...
  46. S

    MHB Finding Eigenvalues, Eigenvectors, [3]

    Consider the system: $x' = x + y + z$ $y' = 0x + 2y + 3z$ $z' = 0x + 0y + 3z$ a)Find the eigenvalues for the systemSo after doing my $3 \times 3$ matrix I got: $\lambda_1 = -3$, $\lambda_2 = 1$, and $\lambda_3 = 2$ , is this correct? b)Find an eigenvector for the smallest eigenvalue So I am...
  47. S

    MHB Solve Eigenvalues, Eigenvectors & General Solution for X'=AX

    Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of **A**. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f)...
  48. R

    Why does this shortcut for eigenvectors of 2x2 symmetric work?

    Hi, I'k looking at some MATLAB code specifically eig2image.m at: http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness-filter/content/FrangiFilter2D So, I understand how the computations are done with respect to the eigenvector / eigenvalues and using...
  49. jk22

    Proof that eigenvalues are constants

    i suppose matrices of the form $$A_i=\left(\begin{array}[cc] ccos(x_i)&sin(x_i)\\sin(x_i)&-cos(x_i)\end{array}\right)$$ And i consider the matrix $$C=A1\otimes A2\otimes 1_4-A1\otimes 1_4\otimes A2$$ I would like to show that the eigenvalues of C are independent of the x i tried with...
  50. S

    MHB Matrices, Eigenvalues and such...

    Consider the system $x'_1 = -5x_1 + 1x_2$ $x'_2 = 4x_1 - 2x_2$ If we write in matrix form as $X' = AX$ then a) X = b) X' = c) A = d) Find the eigenvalues of A. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f) Write the...
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