What is Eigenvectors: Definition and 458 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. jinksys

    Representing |psi> with Continuous Eigenvectors: An Example

    Homework Statement for discrete basis vectors {{e_n}}, a state vector |psi> is represented by a column vector, with elements being psi_n = <e_n|psi>. When basis vectors correspond to those with continuous eigenvalues, vectors are represented by functions. Give such an example of a state...
  2. L

    Eigenspace and basis of eigenvectors

    Homework Statement Given the matrix 0 1 0 0 0 1 -3 -7 -5 Find the eigenspaces for the various eigenvalues Prove that there cannot be a basis of R3 consisting entirely of eigenvectors of AHomework Equations The Attempt at a Solution The...
  3. J

    Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

    Homework Statement i = the 3x3 matrix below 2-λ 0 1 -1 4-λ -1 -1 2 0-λ Using remainder and factor theorem find the 3 values of λ. Homework Equations |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3| |a|=ad-bc The Attempt at a Solution(2-λ) |(4-λ x...
  4. L

    Choosing coefficients for eigenvectors.

    Homework Statement For the following operator represented in the orthonormal basis {|1>, |2>} \hat{M} = \begin{pmatrix} 2 & i\sqrt{2} \\ -i\sqrt{2 & 2} \end{pmatrix} find the eigenvalues and eigenvectors and express them as a function of |1> and |2> normalized.The Attempt at a Solution I'm...
  5. DryRun

    Find the eigenvectors from general equation

    Homework Statement I know the eigenvalues, and i have derived the general equation as follows: -x1 + x2 = 0 x1 - x2 = 0 0.x3 = 0 What is the eigenvector? The attempt at a solution I am trying to find the exact procedure to solve this, but there seems to be none. As a general rule...
  6. L

    Eigenvalues and eigenvectors for linear transformation

    Homework Statement V is a vector space consisting all functions f:R->R that is differentiable many times (a) Let T:V->V be the transformation T(f)=f' Find the (real) eigenvectors and eigenvalues of T (b) Let T be transformation T(f)=f" Prove that all real number, m is the eigenvalue of...
  7. DryRun

    Find the eigenvectors given the eigenvalues

    Homework Statement This is the matrix A, which i need to find the eigenvalues and eigenvectors. 3x3 matrix 5 6 12 0 2 0 -1 -2 -2 The attempt at a solution I have found the eigenvalues to be: 1, 2, 2. So, the final eigenvalues are : 1 and 2. Now, i found the eigenvector for...
  8. T

    Why were eigenvalues and eigenvectors defined?

    I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?
  9. S

    Compute the Eigenvalues and Eigenvectors

    Homework Statement Compute the Eigenvalues and Eigenvectors of A A= [0 0 1;0 2 0;3 0 0] Homework Equations |A-lamda*I|=0 where I know the lamdas and plug them into the above equation and expand the system of equations. The Attempt at a Solution I have solved for the...
  10. P

    Eigenvectors of coupled 2 different mass system

    Homework Statement Solve the coupled mass spring problem for two different masses. Similar to the Shankar example: (d/dt)2x1=-2*w1*x1 + w1*x2 (d/dt)2x2=w2*x1 -2*w2*x2 where w1= k/m1 w2 = k/m2 Homework Equations Eigenvalue problem: UX=uX Diagonalization: A=UDUt Exponential...
  11. D

    Exploring Eigenvectors and Eigenspaces in Linear Transformations

    Homework Statement Let T: M22 - M22 be defined by T(a b = 2c a + c c d) b-2c d (a) Find the eigenvalues T. (b) Find bases for the eigenspaces of T. Homework Equations det(lambdaI - A) = 0 The Attempt at a Solution I just wanted to know how to start here. I am...
  12. A

    Direct Eigenvalue solver, which retains the the order of eigenvectors

    Hi guys I have a problem that I need some help with, I am looking for a direct eigenvalue solver algorithm. The problem is that all the eigenvalue solvers I can find seems to reorder the final matrix after the size of the eigenvalues. The matrix it shall calculate is very small (5-10)...
  13. A

    Eigenvectors of the Hamiltonian

    Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment), So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian: H = A*S_{x}^{2} + B*S_{y}^{2} + C*S_{z}^{2}. Easy enough, just...
  14. B

    Eigenvalues and eigenvectors of a matrix

    Hello i have this matrix \in Z mod 7, M = \begin{pmatrix} 0&6\\ 5&0 \end{pmatrix} always modulo 7 in Z. I found characteristic polynomial x^2+5. Eigenvalues are \lambda = 3, \lambda' = 4 Eigenvectors related to \lambda = 3 are the non-zero solution of the system: 4x +6y = 0, 5x+4y = 0 I...
  15. D

    Non-linearly independent eigenvectors

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  16. Z

    How can I find a basis for the span of some eigenvectors?

    Hello all. This is my first post here. Hope someone can help. Thank you guys in advance. Here is the question: I have a n-by-n matrix A, whose eigenvalues are all real, distinct. And the matrix is positive semi-definite. It has linearly independent eigenvectors V_1...V_n. Now I have known...
  17. R

    Finding eigenvalues and eigenvectors of a matrix A

    Homework Statement Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4] Homework Equations det(A - lambda I) = 0 The Attempt at a Solution After expanding using the formula I have the equation (2 - \lambda)(4 - \lambda) - 8 Which gives \lambda = 0, 6 (Should I...
  18. M

    What are the Eigenvectors of a 3x3 Matrix with Eigenvalues 6 and 2?

    Homework Statement So the 3x3 matrix involved is [3 -1 -1:-4 6 4:-1 1 1], The eigenvalues are L=6 and L=2. Homework Equations (A-LI)e=0 The Attempt at a Solution I stuck the eigenvalues into the matrix and got (-1 1 1)(not sure if its right) for L=2 but when I use L=6 in I can't...
  19. T

    Eigenvectors of rotation matrix

    Homework Statement This question is from Principles of Quantum Mechanics by R. Shankar. Given the operator (matrix) \Omega with eigenvalues e^{i\theta} and e^{-i\theta} , I am told to find the corresponding eigenvectors.Homework Equations \Omega = \left[ \begin{array}{cc} \cos{\theta}...
  20. J

    How can I find the eigenvectors and basis for the eigenspace of a given matrix?

    Homework Statement Im looking at finding the eigenvectors of a matrix but also a basis for the eigenspace A = [ 6 16 ] [ -1 -4 ] lambda = 4 lambda = -2 Homework Equations (A - lambda I ) v = 0 The Attempt at a Solution So with the above equation I get: for...
  21. E

    Find eingenvalues and eigenvectors of an order n matrix

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  22. P

    How to find eigenvectors of ugly transfer matrices in stat mech

    Homework Statement I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix. For example, studying the lattice gas model produces the transfer...
  23. estro

    Why Does My Eigenvector Calculation Differ from WolframAlpha's Result?

    Given the matrix: \left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) I'll find eigenspace for the eigenvalue of t=0, So I have to solve: \left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) {(x,y)}^t=0 Then I do R_2->R_2-2R_1 and get x+2y=0 => x=-2y => get the eigenvector...
  24. M

    MATLAB Matlab eigenvectors of symbolic matrix not working

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  25. I

    Find 2 eigenvectors given an eigenvalue, and find remaining eigenvalue

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  26. A

    Finding eigenvectors of similar matrices

    If v is in Rn and is an eigenvector of matrix A, and P is an invertible matrix, how would you go about finding an eigenvector w of PAP-1? I'm thinking you have to use a fact about similarity?
  27. D

    EigenVectors system of differentials

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  28. P

    On the ‘Sub-Additivity’ of Principal Eigenvectors

    Dear Friends & Colleagues, I have a couple of nagging issues with mathematics I was hoping anyone of you would kindly be able to help resolve. Given a few (let’s say 3) square matrices, A_1, A_2, A_3, and the corresponding principal eigenvector (eigenvector corresponding to the largest...
  29. A

    What are the eigenvectors for lambda = -1?

    Homework Statement find the general solution to x'=Ax; where A is a 3x3matrix: A=[0 1 1; 1 0 1; 1 1 0] Homework Equations det(A-lambda*I)=0 The Attempt at a Solution i found the eigenvalues to be 2, -1, -1. for lambda=2 i found the corresponding eigenvector to be a 3x1 martrix...
  30. S

    Complex Eigenvectors: Showing A(ReV) and A(ImV)

    Homework Statement Let A be a 2x2 matrix with complex eigenvalue (lambda)=a-bi, (b does not equal 0) and associated eigenvector v-> C^2. Show that A(Rev)=aRev +bImv and A(ImV)=-bRev +aImv. Where Re is real part and I am is imaginary part. Homework Equations The Attempt at a...
  31. H

    Eigenvalues and Eigenvectors of Invertible Linear Operators and Matrices

    Let L : V>>>V be an invertible linear operator and let lambda be an eigenvalue of L with associated eigenvector x. a) Show that 1/lambda is an eigenvalue of L^-1 with associated eigenvector x. For this question, the things I know are that L is onto and one to one. Therefore, how to prove this...
  32. Char. Limit

    Linear Algebra practice final - Eigenvectors

    Homework Statement Let \[ A = \begin{array}{ccc} 3 & 5 & 6 \\ 0 & 2 & h \\ 0 & 0 & 2 \\ \end{array} \]. What should be the value of h so that there are two linearly independent eigenvectors of A corresponding to eigenvalue 2? Homework Equations (A-\lambda I) \textbf{x} =...
  33. G

    Meaning of Eigenvectors in a Graph

    What interpretation could the eigenvectors in a graph have? By graph I mean an adjaceny matrix not counting self-loops. If you can draw any physical meanings or point to any examples, that'd be even better. Thank you!
  34. C

    What Are the Mistakes in Multiplying Eigenvectors for Homogeneous Systems?

    In the context of using the eigenvalue method for homogeneous systems, my eigenvector is: V=[1 -i]^T and I have a lambda value of 3-4i. Here's my setup: (e^3t)[1 -i](cos 4t - i sin 4t) For the first equation (real): ME: x1(t) = e^3t(A cos 4t - B sin 4t) BOOK: x(t) = e^3t(A cos 4t + B sin...
  35. S

    Order of Eigenvectors in Matrix Generation: Does it Make a Difference?

    Homework Statement When generating a matrix from eigenvectors, does it matter in which order the columns are placed?
  36. U

    Eigenvectors eigenvalues and constant of motion

    Homework Statement a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = Hx+Hy+Hz, where Hx=1/2*(px2+x2), Hy=1/2*(py2+y2), Hz=1/2*(pz2+z2) Check that the standard angular momentum operators Lx, is a constant of motion. b.) By knowing that the...
  37. G

    How to find eigenvectors of a 3 by 3 matrix

    Question: find the eigenvectors of {{2,-1,-1},{-1,2,-1},{-1,-1,2}}. Equations: none Attempted answer: I have the characteristic equation as x^3-6x^2+9 which gives eigenvalues as 0 3 and 3. This is correct says wolfram alpha for x=0 {{2,-1,-1},{-1,2,-1},{-1,-1,2}}•{x,y,z}=0 I get 3...
  38. T

    Understanding Eigenvectors and Eigenvalues in Linear Algebra

    I am trying to get an eigenvector for the following matrix, I am up to the final step. 4 1 0 0 I got it to be -1 4 is this the same as 1 -4 sorry I am pretty new to linear algebra.
  39. F

    What are Eigenvectors and Eigenvalues?

    Homework Statement http://img820.imageshack.us/img820/4874/cah.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Did it already, 3 is the eigenvalue b) This is just finding the nullspace and the basis of the nullspace are my eigenvectors right? c) ignore...
  40. B

    Order of Eigenvectors in Diagonalization

    I essentially know how to find eigenvalues and thus eigenvectors, though when solving a problem about diagonalization I do not know how to order them (as in, I can find all the eigenvectors but do not know which order to place them into find my X that diagonalizes my A) In the examples of my...
  41. C

    Finding Eigenvectors for a Homogenous System with Three Lambda Values

    I am working on problems involving the eigenvalue method for homogenous systems in my differential equations class. However, I have not had linear algebra yet so I am not understanding some of this. I've made it to the step where I have three lambda values and plugging them back into the...
  42. 2

    What are the Eigenvectors of the Integration Operator with Cosine Kernel?

    Homework Statement Find the eigenvalues and a corresponding system of eigenvectors of the operator Af(x) := Integration from 0 to 1 K(x; y)f(y) dy where K(x; y) = cos (2pi(x - y))
  43. R

    Eigenvectors - why must det(A-λI)=0?

    if we say that an eigenvector u is a vector whose direction is unchanged by the matrix transformation A: Au=λu=λIu rearranging: Au - λIu = 0 u(A-λI)=0 we know the u=0 is obviously a solution. and i also appreciate that if A-λI=0 then A=λI which is just a scalar multiplied by the identity so is...
  44. 0

    Find Eigenvectors of 3x3 Matrix

    Homework Statement Find the eigenvalues and an orthonormal set of eigenvectors for this matrix: |1 1+i 0| |1-i 1 0| |0 0 2| Homework Equations Find the determinant of A - xI, where A is the matrix, I is the identity matrix, and x denotes eigenvalues Set the...
  45. M

    Finding Eigenvectors with a Parameter in Homework Solution?

    Homework Statement Okay, we've got a 3x3 matrix. All the entries are equal to 1. Call the matrix A. Basically It looks like this 1 1 1 1 1 1 1 1 1 find all the eigenvalues and eigenvectors. The Attempt at a Solution Right, I got the eigenvalues being 0 and 3, the same as my...
  46. H

    Find Matrix A from eigenvalues and eigenvectors?

    Homework Statement Matrix A has eigenvalues \lambda1= 2 with corresponding eigenvector v1= (1, 3) and \lambda2= 1 with corresponding eigenvector v2= (2, 7), find A. Homework Equations Definition of eigenvector: Avn=\lambdanvn The Attempt at a Solution I tried this by making...
  47. P

    Square matrix with no eigenvectors?

    Is there such a thing as a square matrix with no eigenvectors? I'm thinking not ... since even if you have: \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] you could just as well say that the eigenvalue(s) are 0 (w/ algebraic multiplicity 2) and the eigenvectors are: u_1 =...
  48. D

    Calculating Eigenvalues and Eigenvectors

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  49. D

    Eigenvalues and Eigenvectors - Find x2(1)

    Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen.jpg The Attempt at a Solution It is a bit too long to type it all out, but I was wondering whether I am correct: I got, A = 7/2 , B = 0 , C = -1/8 , D = 1/8 And from this I worked out, x2(1) =...
  50. F

    Generalized Eigenvectors?

    I see that a generalized eigenvector can be represented as such: (A - λI)xk+1 = xk, where A is a square matrix, x is an eigenvector, λ is the eigenvalue I is the identity matrix. This might be used, for example, if we have duplicate eigenvalues, and can only derive one eigenvector from...
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