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. T

    Help in Finding Eigenvectors Associated with Complet Eigenvalue

    The last matrix at the bottom of the second page is the Eigenvector found using Matlab. I'm trying to find it by hand. I found the Real Eigenvector associated with L=76.2348. But I've tried to find the Eigenvector's for the complex Eigenvalues for a while and can't get the answer given by...
  2. M

    Eigenvalues and eigenvectors of the momentum current density dyadic

    Homework Statement What are the eigenvalues and eigenvectors of the momentum current density dyadic \overleftrightarrow{T} (Maxwell tensor)? Then make use of these eigenvalues in finding the determinant of \overleftrightarrow{T} and the trace of \overleftrightarrow{T}^2 Homework...
  3. S

    What are Eigenkets and Eigenvectors and how do they relate to quantum mechanics?

    Ok I understand how to find eigenvectors, but I don't understand what they are. I am also uneasy with eigenkets and I don't understand what they are also. I need to understand both these topics to get a grasp on quantum mechanics. thank you
  4. D

    Finding the eigenvectors for T()

    Homework Statement Which of the following is not an eigenvector for T \left( \left[ {\begin{array}{cc} x \\ y \\ \end{array} } \right] \right) = \left[ {\begin{array}{cc} x + y \\ x+ y \\ \end{array} } \right] ? A) v = [-2 -2]T B) v = [1 -1]T C) v = [1 2]T D)...
  5. S

    Does Finding an Eigenvalue Guarantee an Eigenvector?

    Is at least one eigenvector guaranteed to exist given that we have found at least one eigenvalue? So, for example, given that we have found an eigenvalue of multiplicity 2 of a matrix, are we guaranteed to find at least 1 eigenvector of that matrix? Why or why not?
  6. I

    How to find eigenvectors of a matrix

    My textbook doesn't seem to explain it clearly enough for me to comprehend. But from what I can see, after getting the eigenvalues, you sub them back into the lambdas that are in the matrix: (\lambda I - A)x = 0 From here, you can solve for the system of equations with Gaussian elimination...
  7. I

    Finding eigenvectors for diagonalization

    Homework Statement Let A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right] Find an invertible S and a diagonal D such that S^{-1}AS=DHomework Equations ...The Attempt at a Solution So first I need to get eigenvalues so I can get the eigenvectors which will give me the...
  8. H

    Eigenvalue and eigenvectors of COMPLEX matrix

    dear all how do you find the eigenvalues and eigenvectors of a complex matrix? 0 ; -i ; 0 ; 0 i ; 0 ; -i*sqrt(2) ; 0 0 ; i*sqrt(2) ; 0 ; -i*sqrt(5) 0 ; 0 ...
  9. T

    Finding eigenvectors of a simple 2x2 matrix

    Homework Statement The matrix is: |1 2| |3 4|The Attempt at a Solution I've worked out the eigenvalues to be \stackrel{\underline{5\pm\sqrt{33}}}{2} But when I plug the first eigenvalue back in I get: |1 - \stackrel{\underline{5+\sqrt{33}}}{2}......2 | |3......4 -...
  10. L

    The Eigenvalues and eigenvectors of a 2x2 matrix

    Homework Statement Let B = (1 1 / -1 1) That is a 2x2 matrix with (1 1) on the first row and (-1 1) on the second. Homework Equations The Attempt at a Solution A) (1 1 / -1 1)(x / y) = L(x / y) L(x / y) - (1 1 / -1 1) (x / y) = (0 / 0) ({L - 1}...
  11. maverick_starstrider

    Finding Eigenvectors with Lanczos Algorithm

    Hi, I'm applying the Lanczos algorithm to find the minimal eigenvalue of some huge matrix. Now that I've got it I'm trying to find the eigenvector corresponding to this eigenvalue. Now I have looked through book after book after book and I have yet to find an explanation of how to do this...
  12. H

    Eigenvectors and diagnolization

    Two questions. First, I'm given a 3x3 matrix with the last row all zeroes. I'm asked to diagonalizable it, but the determinant is 0, so there are no eigenvalues. Am I reasoning correctly here? It seems an odd question to ask. Second, I'm asked to prove that if A n x n matrix in C space, then...
  13. D

    Finding Eigenvectors for 4x4 Matrix A = 4 2, 0 1 | Homework Help

    Homework Statement I have matrix A = 4 2 0 1 Whose eigenvalues I found to be 4 & 1 I need to find the eigenvectors for the same matrix Homework Equations (A-lambdaI)V=0The Attempt at a Solution Lambda = 4 gives 0 2 x V1 = 0 0 -3 V2 0v1 + 2v2 = 0 0v1 - 3v2 = 0...
  14. B

    Calculating eigenvectors of complex numbers

    Homework Statement Find the eigenvectors of the matrix A Homework Equations 3. The attempt at the solution \[ \left( \begin{array}{cc}4 & -5 \\1 & 0 \end{array} \right)\] First I find the characteristic equation A - \lambda I \[ \left( \begin{array}{cc}4-\lambda & -5...
  15. C

    Trace of a matrix and an expansion of eigenvectors

    Hi, I'm trying to derive the Kullback-Leibler divergence between two multi-variate gaussian distributions, and I need the following property. Is there a simple way to understand this? Prove that: Given that E has orthonormal eigenvectors u_{i} and eigenvalues \lambda_{i} Then: trace(A*E) =...
  16. E

    Eigenvalue with multiplicity k resulting in k orthogonal eigenvectors?

    I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me. "Properties of the eigenvalue when A is symmetric. If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times), orthogonal...
  17. B

    Using Eigenvalues and Eigenvectors to solve Differential Equations

    Homework Statement x1(t) and x2(t) are functions of t which are solutions of the system of differential equations x(dot)1 = 4x1 + 3x2 x(dot)2 = -6x1 - 5x2 Express x1(t) and x2(t) in terms of the exponential function, given that x1(0) = 1 and x2(0) = 0 The Attempt at a Solution I've already...
  18. S

    Eigenvalues and eigenvectors of this matrix

    Consider the nXn matrix A whose elements are given by, A_{ij} = 1 if i=j+1 or i=j-1 or i=1,j=n or i=n,j=1 = 0 otherwise What are the eigenvalues and normalized eigenvectors of A??
  19. Z

    Eigenvalues and Eigenvectors uniquely define a matrix

    Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix M from a matrix of its eigenvectors as columns P and a diagonal matrix of the eigenvalues E through M=P E P^{\dagger}?
  20. F

    What are the eigenvalues and eigenvectors of matrix A = [2 2; 3 1]?

    Eigenvalues & Eigenvectors !SOLVED! Homework Statement Find the eigenvalues and eigenvectors of matrix A = \left( \begin{array}{cc} 2 & 2 \\ 3 & 1 \end{array} \right) Homework Equations Ax = \lambda x The Attempt at a Solution Solving \left\vert \begin{array}{cc} 2 - \lambda &...
  21. E

    Show that V has a basis of eigenvectors

    Homework Statement Let T: V---->V be a linear operator where dim V=n. Show that V has a basis of eigenvectors if and only if V has a basis B such that TB is diagonal.Homework Equations The Attempt at a Solution Let T=[a1,1...an,1] ai,j=/=0 [a1,n...an,n]...
  22. malawi_glenn

    Eigenvalues and eigenvectors of symmetric 2x2 matrix?

    Hello I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0. I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me? cheers
  23. 9

    Eigenvalues and Eigenvectors of 3x3 matricies

    Hello Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues. Can someone please help.. Thanks
  24. J

    What are the Eigenvalues and Eigenvectors of Similar Matrices

    Homework Statement Let A and B be similar matrices a)Prove that A and B have the same eigenvalues Homework Equations None The Attempt at a Solution Firstly, i don't see how this can even be possible unless the matrices are exactly the same :S
  25. J

    Eigenvectors of first-order differential equation

    Suppose that f is an eigenvector of T with corresponding eigenvalue \lambda. Then f' = T(f) = \lambdaf. This is a first-order differential equation whose solutions are of the form f(t) = ce^(\lambda*t) for some scalar constant c. Consequently, every real number \lambda is an eigenvalue of T...
  26. A

    Linear independence of eigenvectors

    How can you show that an arbitrary n \times n matrix has n linearly independent eigenvectors? What if all you know about the matrix is that it's the product of a positive-definite matrix and a semi-positive-definite matrix?
  27. S

    Eigenvectors and Manipulations on the Matrix

    If x is an eigenvector of matrix A, is it true that it is also an eigenvector of A -1, or A + A^2? Thanks for the help.
  28. S

    Matrix Eigenvectors: How Many Are Linearly Independent?

    Hi, I am a little confused how do you find out when a matrix has two independent eigenvectors or when it has one or when it has more than two, or is it possible it can have no eigenvectors.
  29. S

    What are the unit eigenvectors for the matrix A = [5 -2; -2 8]?

    Homework Statement I'm trying to find the unit eigenvectors corresponding to the following matrix A = [5 -2; -2 8] ; means new row Homework Equations det(A - hI) = 0 The Attempt at a Solution I get lambda = 4 and 9 unit eigenvector corresponding to lambda = 4 x1 = ( -2/sqrt(5)...
  30. J

    Solving a Matrix Equation: Decoupling and Eigenvectors

    Hello, please can someone tell me how to decouple and solve this equation? It was on a problem sheet, but the solution jumped to the decoupled equation... =( \frac{dx}{dt} = 2x+y-t \frac{dy}{dt}=2x-y+t I know that it can rewritten as \frac{d}{dt}\left[ \begin{array}{cccc} 2 & 1\\...
  31. S

    Can I choose any number for the variable when finding eigenvectors in matrices?

    When finding eigenvectors in matrices I choose something for some x-es. Like sometimes x3 or x4 is chosen to be s or t or 2s etc... What I´d like to ask about is, does it not matter what the number is? Can I chose whatever I want to? If the matrix has 3 eigenvalues and after gauss...
  32. R

    -4+2y=0 and y=2x comes from Eigenvectors

    I don't understand where -4+2y=0 and y=2x comes from Is it obtained after finding the determinant or are the equations reconstructed that the matrix was created from? http://users.on.net/~rohanlal/eigen1.jpg I also don't understand what's happening here. Where did the x(1,2) come from and...
  33. S

    How to find eigenvectors of 2x2 by gauss jordan method

    Homework Statement how to find eigenvectors by using gauss jordan A=[1 1; 2 2] Homework Equations I know how to use gauss jordan but don´t know how to use it to find eigenvectors The Attempt at a Solution First I find the eigenvalues: ((y-1)(y-2)-(1*2)=> y1=0 and y2=3 Then I...
  34. C

    Finding Eigenvectors with Close Eigenvalues

    Homework Statement Given the characteristic polynomial -2+x-2x^2-x^3, find the eigenvalues and eigenvectors of the matrix [-1, -1, 0] [1, 1, 1] [3, 1, -2] Homework Equations The Attempt at a Solution The eigenvalues are -2.659, 0.329-.802i, and 0.329+.802i. Next you plug each eigenvalue into...
  35. F

    Eigenvalues and eigenvectors

    my question is take A= {(5,0,-1),(2,3,-1),(4,0,1)} find all eigenvalues and eigenvectors by using the characteristic equation i get -(lamda-3)3 however its the next bit i don't understand, in the answers (A-3I)(x,y,z)=(0,0,0) is used which I'm perfectly ok with and then (A-3I)2 is used and...
  36. M

    Treating operators with continuous spectra as if they had actual eigenvectors?

    I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...
  37. R

    Edu - "How to Find Eigenvectors of a Matrix: A Step-by-Step Guide

    1. Find the eigenvalues and the eigenvectors corresponding to eigenvalues of the matrix A = \left[\begin{array}{ccccc} 1 & 3 \\ 4 & 2 \end{array}\right] 3. The Attempt at a Solution (\lambda I - A) = \lambda \left[\begin{array}{ccccc} 1 & 0 \\ 0 & 1 \end{array}\right] -...
  38. B

    Finding Eigenvectors for a Matrix A

    Homework Statement a matrix A: [1 3 0 3 1 0 0 0 -2] Find Q and D where QTAQ=D The Attempt at a Solution I found the eigenvalues of -4,2,2 When I plug them back in and rref the matrix I only get the trivial solution meaning the matrices are linearly independent. How do I get...
  39. F

    How can generalized eigenvectors help with finding a matrix in Jordan form?

    revising for a text and got stuck mid way through a question Find the eigenvalues and vectors of A (in matrix form i will state colum then column then column) A((3,0,0),(0,2,0),(0,0,2) B=((3,0,0),(0,2,1),(0,0,2) for A i got x(lamda)=(lamda-3)(lamda-2)^2 lamda=2or3 then i got lamda=3 solved...
  40. A

    Eigenvectors of commuting matrices

    I can't follow an argument in Horn and Johnson's Matrix analysis in a suggestion (actually an outline of a proof) that follows problem 8 following section 1.3 (pg 55 in my copy). They argue that if A and B are complex square matrices of order n which commute, and if all eigenvalues of B are...
  41. L

    Strange question regarding eigenvectors / eigenvalues

    Homework Statement Suppose that the 2x2 matrix A has eigenvalues lambda = 1,3 with corresponding eigenvectors [2,-1]^T and [3,2]^T. Find a formula for the entries of A^n for any integer n. And then, find A and A^-1 from your formula. Homework Equations Ax = lambda X (P^-1)AP = D A =...
  42. U

    Complex Eigenvectors, How do I check?

    I have come to a problem where I have Eigenvalues = 2,2i,-2i and my Eigenvectors have i's in them. I usually check my work using my calculator to perform the operation of, S^{-1}AS=J where S is my Eigenvector matrix, A is my original. I then see what my J matrix looks like. It should...
  43. K

    What does it mean to find eigenvectors of some operator in some basis?

    I am studying QM by myself. I got a quite confusing problem which annoying me for a certain time. Well, this question is about the angular momentum opeator Lx, Ly and Lz. The matrix form for these operatore are given, so by solving the corresponding secular equation, it is easy to find the...
  44. D

    Geometric Properties from Eigenvectors

    Homework Statement Hi! I just used MATLAB to find the eigenvalues and eigenvectors of A=[0 -1; 1 0] I obtained the eigenvalues of 0 +/- i and eigenvectors of v(1) = [ 0.7071; 0 - 0.7071i] and v(2) = [ 0.7071; 0 + 0.7071i] Homework Equations I'm having trouble interpreting these results in...
  45. B

    Find 2 linearly independent eigenvectors and a eigenvalue

    Hi I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A 5 5 5 5 5 5 5 5 5 The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there...
  46. Somefantastik

    Finding the eigenvectors (and behavior of solution) around the

    finding the eigenvectors (and behavior of solution) around the critical points found in this thread: https://www.physicsforums.com/showthread.php?t=258349&referrerid=110346 D_{f} = \[\begin{pmatrix}32x & 18y \\ 32x & -32y\end{pmatrix}\] D_{f}(1,1) = \[\begin{pmatrix}32 & 18 \\ 32 &...
  47. E

    Principal Component Analysis: eigenvectors?

    Hey Hello, I am dealing with som Principal Component Analysis Can anyone explain why the first eigenvector of a covariance matrix gives the direction of maximum variability. why this special property of eigenvectors
  48. D

    Why do we need generalized eigenvectors for matrices with repeated eigenvalues?

    So I understand that if an nxn matrix has n distinct eigenvalues that you can diagonalize the matrix into S\LambdaS^{-1}. This is important because then this form has lots of good properties (easy to raise to powers, etc) So when there are not n distinct eigenvalues, you then solve...
  49. S

    Eigenvalues & eigenvectors of N x N matrix?

    How to get eigenvalues & eigenvectors of N x N matrix? Please can anyone help me out i have searched a lot but not able to find the solution. Regards
  50. F

    Linear Algebra Question: Eigenvectors & Eignvalues

    This is my last week in Linear Algebra. I am working on our last homework assignment before the exam so I want to make sure I know what I am doing. In each part, make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering...
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