What is Eigenvector: Definition and 148 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. N

    Unique Eigenvector of a: Unveiling Coherent States

    Homework Statement Show that, for all complexe numbers \alpha, a has a unique eigenvector |\alpha\rangle that is nothing else but the coherent state \psi(x)=\frac{e^{-\frac{i}{2\hbar}\langle X\rangle\langle P\rangle}}{(\pi\ell^2)^{1/4}}e^{-\frac{(X-\langle...
  2. R

    Solving Zero Eigenvector: A Homework Problem

    Homework Statement I can calculate the proper eigenvalues, but when I plug them back into the matrix, I get x1=0 and x2=0. But this is not the answer Maple gives me! How do I solve for the eigenvector when it appears that a zero vector is the only solution? Homework Equations For...
  3. A

    Eigenvector with Complex Eigenvalues - What am I doing wrong?

    Homework Statement Homework Equations Conjugate of a complex number Matrix reductionThe Attempt at a Solution My attempt is bordered. Sorry about the quality. So I'm not sure what I'm missing. I use the exact same method that I use for normal eigenvectors, just with complex numbers in the mix.
  4. W

    Proof: Eigenvector of B Belonging to \lambda for A*S*x

    Homework Statement Let B = S^-1 * A * S and x be an eigenvector of B belonging to an eigenvalue \lambda. Show S*x is an eigenvector of A belonging to \lambda. Homework Equations The Attempt at a Solution The only place I can think of to start, is that B*x = \lambda*x. However...
  5. I

    When to use eigenvector method

    I'm a physics major. As such, I have come across several situations in my studies that require the use of eigenvectors and eigenvalues. Whenever I have to use this method, I've been told to. I do not have a complete understanding of eigenvectors and values and am wondering how you would spot a...
  6. Z

    Linear Algebra - Eigenvector question

    Homework Statement The linear operator T on R2 has the matrix \begin{bmatrix}4&-5\\-4&-3 \end{bmatrix} relative to the basis {(1,2), (0,1)} Find the eigenvalues of T, and obtain an eigenvector corresponding to each eigenvalue...
  7. N

    2x2 matrix A has only one eigenvalue λ with eigenvector v

    This is a revision problem I have come across, I have completed the first few parts of it, but this is the last section and it seems entirely unrelated to the rest of the problem, and I can't get my head around it! Suppose that the 2x2 matrix A has only one eigenvalue λ with eigenvector v...
  8. W

    Understanding Eigenvalues and Eigenvectors in Reflection Matrices

    Homework Statement Let A be a matrix corresponding to reflection in 2 dimensions across the line generated by a vector v . Check all true statements: A. lambda =1 is an eigenvalue for A B. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda =1. C...
  9. F

    Diagonalization, which eigenvector is found?

    Hi! This might be a silly question, but I can't seem to figure it out and have not found any remarks on it in the literature. When diagonalizing an NxN matrix A, we solve the characteristic equation: Det(A - mI) = 0 which gives us the N eigenvalues m. Then, to find the eigenvectors v...
  10. G

    Solve Eigenvector Problem: Find Eigenvalues & Eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors of \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 1 & 1 & 0 \end{array} \right) Homework Equations p(\lambda) = det(A - \lambda I) = 0 The Attempt at a Solution A - \lambda I = \left( \begin{array}{ccc} 2-\lambda & 0 & 0 \\ 0...
  11. F

    What Is the Variance of an Eigenvector in a Hermitian Operator?

    Homework Statement On page 5 of http://arcsecond.wordpress.com/2009/08/01/the-cauchy-schwarz-inequality-and-heisenbergs-uncertainty-principle/ the author states (w/o proof) that if \psi is an eigenvector (say with eigenvalue \lambda) of an Hermitian operator A (I don't think the Hermitian-ness...
  12. J

    Numerical algorithms for finding an eigenvector

    All matrices A\in\mathbb{C}^{n\times n} have at least one eigenvector z\in\mathbb{C}^n. I'm interested to know what kind of algorithms there are for the purpose of finding an eigenvector. I noticed that \frac{|z^{\dagger} A z|}{\|Az\|} = 1\quad\quad\quad\quad (1) holds only when z is an...
  13. E

    Prove Eigenvectors Linearly Independent: v & w

    Homework Statement If v and w are eigenvectors with different (nonzero) eigenvalues, prove that they are linearly independent. Homework Equations The Attempt at a Solution Define an operator A such that a is an nxn matrix, and Av=cIv with c an eigenvalue and v and eigenvector. Define a basis...
  14. J

    Solving Eigenvector Problems: A+B and AB with Corresponding Eigenvalues

    Homework Statement Suppose that v is an eigenvector of both A and B with corresponding eigenvalues lambda and mui respectively. Show that v is an eigenvector of A+B and of AB and determine the corresponding eigenvalues Homework Equations The Attempt at a Solution Av = lambda*v Bv...
  15. C

    Symmetric Matrix Eigenvector Proof

    Eigenvalue and eigenvector for a symmetric matrix Homework Statement Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A...
  16. A

    Solving Ax=b, when b is an eigenvector

    Let A be a nonsingular matrix. What can you say about the convergence of GMRES to the solution of Ax=b when b is an eigenvector of A? I know that if A is a nonsingular matrix with minimal polynomial of degree m, then GMRES solves Ax=b in at most m iterations. Because b is an eigenvector...
  17. D

    Finding eigenvector from eigenvalue

    Homework Statement For the matrix A = -1, 5 -2, -3 I found the eigenvalues to be -2 + 3i and -2 - 3i. Now I need some help to find the eigenvectors corresponding to each. Homework Equations The Attempt at a Solution For r = -2 + 3i, I plugged that into the (A - Ir) matrix...
  18. J

    Need to find eigenvector that corresponds to max eigenvalue

    To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue. Let's say I already have some method that gives me all the eigenvectors and all...
  19. F

    Eigenvector of A_n: Show & Find Eigenvalue

    Homework Statement Directly show that the n x 1 vector [1 1 1 ...1]T is an eigenvector of An. What is its associated eigenvalues? Homework Equations N/A The Attempt at a Solution I started going over this topic since we did not cover it in class due to time constraints and I do not know...
  20. D

    Eigenvectors of Matrix: Solving Basics

    Homework Statement Find the eigenvectors of the following matrix: \[ \left( \begin{array}{ccc} 1 & 1 \\ 4 & -2 \end{array} \right)\] Homework Equations N/A The Attempt at a Solution I already know how to find the solutions. They are {1 1} and {-1 4}. My question is this: could a...
  21. S

    Solve Eigenvector Equation: Prove Roots are Scalars

    Hi, this is actually for my general relativity class, but I thought I would get more help in the math section of the forums, since it involves very little physics, or even not at all. Homework Statement Take Tab and Sab to be the covariant components of two tensors. Consider the determinant...
  22. X

    Finding a normalized eigenvector

    ok, i know how to find an eigenvalue and an eigenvector that's fine, what i don't remember is how to normalize your eigenvectors in my problem i have 2 eigenvectors, (1,3) and (3,1) (1,3) corresponds to eigenvalue 10 (3,1) corresponds to eigenvalue 20 in my notes i have written 'to...
  23. S

    How to Compute Eigenvectors for Large Matrices Efficiently?

    Hi, this is my first post here, so bare with me. So I need to compute the eigenvectors of a large matrix (1000x1000) to (10000x10000x) or so. However, I already have the eigenvalues and diagonal/superdiagonal form of the matrix. The equation (A-lambda*I)*v = 0, where A is the matrix, lambda...
  24. C

    Transforms that Preserve The Dominant Eigenvector?

    Hi, I'm working with stochastic matrices (square matrices where each entry is a probability of moving to a different state in a Markov chain) and I am looking for transforms that would preserve the dominant eigenvector (the "stationary distribution" of the chain). What I want to do is to...
  25. rocomath

    Orthogonal Eigenvector, Proof is bothering me

    Suppose A\overrightarrow{x}=\lambda_1\overrightarrow{x} A\overrightarrow{y}=\lambda_2\overrightarrow{y} A=A^T Take dot products of the first equation with \overrightarrow{y} and second with \overrightarrow{x} ME 1) (A\overrightarrow{x})\cdot...
  26. K

    Finding the Steady State Eigenvector for Flower Mixture

    Homework Statement In order to prodice a more balanced, sustainable long term mix of flower Skwhere the percentage of the offspring of red flowers are p pink and 1-p redSk= A SoSk= [r p w]Twhere A=[1-p* 1/4* 0p***** 1/2* 1/20***** 1/4* 1/2]determine the stady state eigenvectors(you do not have...
  27. R

    Find Corresponding Eigenvectors for Matrices A and B | Quick Help

    Homework Statement The matrix,A,given by A = \left( \begin{array}{ccc} 7 & -4 & 6\\ 2 & 2 & 2 \\ -3 & 4 & -2 \ \end{array} \right) has eigenvalues 1,2,4 . Find a set of corresponding eigenvectors. Hence find the eigenvalues of B, where B = \left( \begin{array}{ccc}...
  28. H

    Solving Complex Eigenvector for (-1 + i \sqrt{11})

    [SOLVED] Complex Eigenvector I need to solve for an eigenvector using the complex eigenvalue -1 + i \sqrt{11} . I have a matrix: A = \left(\begin{array}{cc}-3 & -5 \\3 & 1\end{array}\right) From the equation A \vec{V} = \lambda \vec{V} , where \vec{V} = (x, y) I get : -3x - 5y =...
  29. R

    Normalizing eigenvector with complex entries

    Homework Statement Hi, I'm having a bit of a problem normalizing eigenvectors with complex entries. Currently the eigenvector I'm looking at is \[\vec{v}= \left(\begin{array}{c} -2+i\\ 1 \end{array}\right)\] Homework Equations The Attempt at a Solution If the eigenvectors...
  30. J

    Obtain an eigenvector corresponding to each eigenvalue

    Homework Statement The linear operator T on R^2 has the matrix [4 -5; -4 3] relative to the basis { (1,2), (0,1) } Find the eigenvalues of T. Obtain an eigenvector corresponding to each eigenvalue.Homework Equations The Attempt at a Solution I was able to find the eigenvalues (8 and -1)...
  31. M

    Given a Hamiltonian, find eigenvalues and eigenvector

    Problem : in the Attach file
  32. J

    Eigenvector existence in complex space

    I'm reading a proof where there's a conclusion: "Since zW\subset W, there is an eigenvector v\neq 0 of z in W, zv=\lambda v." There W is a subspace of some vector space V, and z is a matrix, in fact a member of some solvable Lie algebra \mathfrak{g}\subset\mathfrak{gl}(V). (Could be irrelevant...
  33. Q

    Eigenvector Help: Solving Problems 3 & 4

    http://orion.math.iastate.edu/vika/cal3_files/lec33267.pdf i searched eigenvectors on google and this showed up. here are some problems i need further explaining for example 3 and 4. 3. where do they get e1+e2= 0 equation from? then where did they get e= (1,-1) 4. where do they...
  34. S

    How Do We Calculate Eigenvalues for Different Matrices?

    Hi Guys, I have got some enquires for eigenvalue and eigenvector. Consider the 1st matrix: A = [ 1 2 3] [ 0 5 6] [ 0 6 5] The characteristic polynomial is det(A-λI) = [ 1-λ 2 3] [ 0 5-λ 6] [ 0 6...
  35. M

    What should I do when faced with a mixture of real and complex eigenvalues?

    I need to solve the differential equation \mathbf{x'} = \left( \begin{array}{ccc} 3 & 0 & -1\\ 0 & -3 & -1\\ 0 & 2 & -1 \end{array} \right) \mathbf{x} solving for the eigenvalues by taking the determinate and using the "basketweave" yields (3 - \lambda)(-3-\lambda)(-1-\lambda) +...
  36. M

    Trouble with complex eigenvector

    The problem is to solve the differential equation where \mathbf{x'} = \left( \begin{array}{cc} 1 & -5\\ 1 & -3 \end{array} \right) \mathbf{x} given that \mathbf{x(0)} = \left( \begin{array}{cc} 5 \\ 4 \end{array} \right) The eigenvalues are easy to find, and they are: \lambda...
  37. B

    Eigenvector Question: Is v=(2,1) the same as v=(4,2)?

    If I find a given eigenvector , that vector spans the entire eigenspace defined by that eigenvalue correct? Let's say I get v=(2,1) as an eigenvector. That is the same as saying v=(4,2) right? since they are spanning the same space?
  38. G

    Constructing Eigenvectors from Commuting Matrices: A Unique Classification

    Hey all, I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H? From these pairs of eigenvalue is it possible to obtain the eigenvectors? I don't quite know how to procede any suggestions? Thanks...
  39. M

    Eigenvalue and Eigenvector problem

    Hi Given a 3x3 matrix A = \[ \left[ \begin{array}{ccc} 0 & 0 & 1+2i \\ 0 & 5 & 0 \\ 1-2i & 0 & 4 \end{array} \right] I need to a another 3x3 which satisfacies D = U^-1 A U Step 1. Finding the eigenvalues 0 = det(A- \lambda I ) = (0- \lambda)(\lambda - 5) (\lambda -4...
  40. mattmns

    Lin Alg - Eigenvector Existence proof

    Lin Alg - Eigenvector Existence proof (More of a proof ?, than eigenvector ?) Here is the question from my book: Show that If \theta \in \mathbb{R}, then the matrix A = \left(\begin{array}{cc}cos \theta & sin \theta \\ sin \theta & -cos \theta \end{array}\right) always has an...
  41. S

    How do I find the eigenvectors for matrices V & T with known eigenvalues?

    Hi, I've got these two matrices (V & T) and omega square, which is what I have found to be the eigenvalues. Could anyone tell me if this is the way to find the eigenvectors for these matrices and if they are correct? Thanks... http://img305.imageshack.us/img305/6937/ok4zd.jpg
  42. J

    Solve for Eigenvector: Characteristic Equation of Matrix A with Solution Method

    hey, not sure how this works on this website but was just wondering if someone can tell me if I'm doing this right... find characteristic equation of matrix A=l 4 0 1 l l -2 1 0l l -2 0 1l i found it to be (L=lambda)... L^3 - 6L^2 + 11L - 6 = 0 only because i solved...
  43. B

    Verify that a vector is an eigenvector of a matrix

    Hi could someone explain to me how to verify that a vector is an eigenvector of a matrix without explicitly carrying out the calculations which give the eigenvalues of the matrix? Here is an example to illustrate my problem. Q. Let M = \left[ {\begin{array}{*{20}c} { - 3} & 1 & { - 2} & 4...
  44. M

    Solving Eigenvector Problem

    Hi all, I have the following question. A = nxn non singular matrix I = nxn identity matrix li = eigevalues of A i=1,2...n ui = eigenvectors corresponding to the previous eigenvalues. It true that ( A - l1 * I ) * x =0 is satisfied by any vector of the form x = a1 * u1...
  45. A

    Does the notation used for the span part of an eigenvector matter?

    If you have the following kernel (I think that's what it's called): A-\lambda I=\begin{pmatrix}4 & 1 \\ 4 & 1\end{pmatrix} You could write the eigenvector as: \operatorname{span}\begin{pmatrix}1 \\ -4\end{pmatrix} My question is: does it matter how you write the "span" part of it...
  46. M

    Eigenvectors of a 3x3 Matrix A: Calculation and Verification

    Hi I have this here matrix A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 3 & 3 & 0 \end{array} \right] I calculate the eigenvalues and get (2,1,-1) Next I calculate the eigenvectors and get (1,0,1) and (-1,1,0) and (0,0,0) My professor says my third eigenvector is...
  47. D

    Eigenvector Algor differences?

    Hi, Does different eigenvector algorithm give different result? eg. using QL with implicit shifts frm (Numerical Recipes) vs Matlab's LAPACK routines? or anyone knows what method Matlab's LAPACK uses & where i can find the source code in c++? Are eigenvectors unique? Thanks!
  48. S

    Eigenvalue and Eigenvector

    can anyone explain the the real meaning and purpose of eigen vlaue and eigen vectors.. :smile:
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