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. 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...
  2. 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...
  3. P

    Eigendecomposition using cuSolver

    I am looking for a clear example on how to obtain the complete set of eigenvalues and eigenvectors for a dense, non-hermitian matrix using cuSolver.
  4. 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?
  5. 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...
  6. B

    Mohr's circle and formula for eigenvectors

    Don't exist formula for the eigenvectors, all right!? Eigenvectors needs be found manually, correct!? But and about the Mohr's circle? This physical/mathematical theory don't define clearly the direction of the eigenvectors (called principal direction) with the eigenvalues (called principal...
  7. 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...
  8. B

    MHB Eigenvectors of 2*2 rotation matrix

    Hi Folks, I calculate the eigenvalues of \begin{bmatrix}\cos \theta& \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix} to be \lambda_1=e^{i \theta} and \lambda_2=e^{-i \theta} for \lambda_1=e^{i \theta}=\cos \theta + i \sin \theta I calculate the eigenvector via A \lambda = \lambda V as...
  9. B

    Is matrix hermitian and its eigenvectors orthogonal?

    I calculate 1) ##\Omega= \begin{bmatrix} 1 & 3 &1 \\ 0 & 2 &0 \\ 0& 1 & 4 \end{bmatrix}## as not Hermtian since ##\Omega\ne\Omega^{\dagger}## where##\Omega^{\dagger}=(\Omega^T)^*## 2) ##\Omega\Omega^{T}\ne I## implies eigenvectors are not orthogonal. Is this correct?
  10. C

    Are Both Eigenvectors Correct?

    say for example when I calculate an eigenvector for a particular eigenvalue and get something like \begin{bmatrix} 1\\ \frac{1}{3} \end{bmatrix} but the answers on the book are \begin{bmatrix} 3\\ 1 \end{bmatrix} Would my answers still be considered correct?
  11. B

    MHB Finding eigenvectors for a double root

    I have the following matrix: [-1 3 -3(2)^0.5 ; 3 -1 -3(2)^0.5 ; -3(2)^0.5 -3(2)^0.5 2] I was able to find the eigenvalues as -4 and 8. I am now trying to find the corresponding eigenvectors how since -4 is a double root i am unsure how to go about this. I have tried using gaussian...
  12. 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...
  13. 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...
  14. 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...
  15. 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...
  16. 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)...
  17. P

    Find two linearly independent eigenvectors for eigenvalue 1

    Homework Statement A linear transformation with Matrix A = ## \begin{pmatrix} 5&4&2\\ 4&5&2\\ 2&2&2 \end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1. Homework Equations 3. The Attempt at a Solution [/B] I know from the...
  18. 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...
  19. jdawg

    Solve Matlab Eigenvectors Homework

    Homework Statement I think this problem is supposed to be pretty simple, but I have almost no knowledge of how to use matlab. I was told to use this function: [V,D]=eig(A) to give me the eigenvectors (columns of matrix V) and the diagonal matrix with eigenvales in the diagonal ( matrix D). I...
  20. D

    Eigenvectors of Inertia tensor

    Hi, I've written a little fortran code that computes the three Eigenvectors \vec{v}_1, \vec{v}_2, \vec{v}_3 of the inertia tensor of a N-Particle system. Now I observed something that I cannot explain analytically: Assume the position vector \vec{r}_i of each particle to be given with respect...
  21. nomadreid

    Zero eigenvalues or eigenvectors

    I have a bit of problem with zero eigenvectors and zero eigenvalues. On one hand, there seems to be nothing in the definition that forbids them, and they even seem necessary to allow because an eigenvalue can serve as a measurement and zero can be a measurement, and if there is a zero eigenvalue...
  22. M

    On the orthogonality of Sturm-Liouville eigenvectors

    From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...
  23. M

    On the orthogonality of Sturm-Liouville eigenvectors

    From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...
  24. L

    Unitary operator eigenvectors

    Homework Statement I know that Unitary operators act similar to hermitean operators. I want to prove that the eigenvalues of unitary operators are complex numbers of modulus 1, and that Unitary operators produce orthogonal eigenvectors. Homework Equations U†U = I U-1=U† λ = eiΦ{/SUP]...
  25. P

    Eigenvectors of exponential matrix (pauli matrix)

    Homework Statement Find the eigenvectors and eigenvalues of exp(iπσx/2) where σx is the x pauli matrix: 10 01 Homework Equations I know that σxn = σx for odd n I also know that σxn is for even n: 01 10 I also know that the exponential of a matrix is defined as Σ(1/n!)xn where the sum runs...
  26. Clear Mind

    Complete set of eigenvectors question

    Suppose you have two observables ##\xi## and ##\eta## so that ##[\xi,\eta]=0##, i know that there exists a simultaneous complete set of eigenvectors which make my two observables diagonal. Now the question is, if ##\xi## is a degenerate observable the complete set of eigenvectors still exist?
  27. T

    Abstract Linear Algebra: Eigenvalues & Eigenvectors

    Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...
  28. D

    Differential Equations System Solutions

    Homework Statement Consider the initial value problem for the system of first-order differential equations y_1' = -2y_2+1, y_1(0)=2 y_2' = -8y_1+2, y_2(0)=-1 If the matrix [ 0 -2 -8 0 ] has eigenvalues and eigenvectors L_1= -4 V_1= [ 1...
  29. S

    More than 3 eigenvectors perpendicluar

    all the eigenvectors of a matrix are perpendicular, ie. at right angles to each other, HOW? I can imagine three eigenvectors as three perpendicular axes. How can be more than three axes are perpendicular with respect to each other?
  30. A

    Comp Sci Eigenvalues and eigenvectors of a real symmetric matrix in Fortran

    Homework Statement I try to run this program, but there are still some errors, please help me to solve this problems Homework EquationsThe Attempt at a Solution Program Main !==================================================================== ! eigenvalues and eigenvectors of a real...
  31. DavideGenoa

    Eigenvectors of Fourier transform operator #F:L^2\to L^2#

    Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...
  32. H

    Number of eigenvectors for Hermitian matrices

    Hello, I am currently trying to study the mathematics of quantum mechanics. Today I cam across the theorem that says that a Hermitian matrix of dimensionality ##n## will always have ##n## independent eigenvectors/eigenvalues. And my goal is to prove this. I haven't taken any linear algebra...
  33. C

    Eigenvectors of decomposed matrix

    Hello everyone, I've got the eigenvectors of a matrix H (Hessenberg matrix) obtained from the decomposition A=QHQ'. Now I seek the eigenvectors of the matrix A. I've found somewhere that it should be : eigenvector_of_A=Q*eigenvector_of_H but some numerical test with MATLAB doesn't agree. For...
  34. J

    Function scales eigenvalues, but what happens to eigenvectors?

    Statement: I can prove that if I apply a function to my matrix (lets call it) "A"...whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers...
  35. A

    Eigenvalues and Eigenvectors of a Hermitian operator

    Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...
  36. E

    Finding eigenvectors of 3x3 matrix

    Hi, I have an issue with zeroing the 3x3 matrix to find the eigenvector. I have found the characteristic equation for the 3 eigenvalues. the matrix is 1 1/2 1/3 1/2 1/3 1/4 1/3 1/4 1/5 The equation i got is -A^3 + (23/15)A^2 - (127/720)A + (1/2160) which...
  37. B

    Computing Eigenvectors of Matrix H with Wolfram

    Homework Statement Compute, using wolfram is an option, eitH where H is: ##H = \begin{pmatrix} 0 & -1 & 0 & 0 & 0& 0\\ -1 & 0 & -1 & 0 & 0& 0\\ 0 & -1 & 0 & -1 & 0& 0\\ 0 & 0 & -1 & 0 & -1& 0\\ 0 & 0 & 0 & -1 & 0& -1\\ 0 & 0 & 0 & 0 & -1& 0\\ \end{pmatrix}## Homework...
  38. P

    Eigenvalues and eigenvectors of a non-symmetric matrix?

    I have a non symmetric matrix AB where A and B are symmetric matrices. How can I find the eigenvectors and eigenvalues of AB? In a paper( Fisher Linear Discriminant Analysis by M Welling), the author asks to find eigenvalues and eigenvectors of B^(1/2)* A *B^(1/2) which is a symmetric...
  39. ShayanJ

    Eigenvectors of this Hamiltonian

    I've got a problem which is asking for the eigenvalues and eigenstates of the Hamiltonian H_0=-B_0(a_1 \sigma_z^{(1)}+a_2 \sigma_z^{(2)}) for a system consisting of two spin half particles in the magnetic field \vec{B}=B_0 \hat z . But I think the problem is wrong and no eigenstate and...
  40. J

    Eigenvalues / Eigenvectors relationship to Matrix Entries Values

    Hi, folks I have had a hard time to find out whether or not there is a theorem in Linear Algebra or Spectral Theory that makes any strong statement about the relationship between the entries of a Matrix and its Eigenvalues and Eigenvectors. Indeed, I would like to know how is the...
  41. P

    Proving a property of eigenvalues and their eigenvectors.

    Homework Statement I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of A + kI. The Attempt at a Solution ## A\vec{v}=\lambda\vec{v} ## ## (A+kI)\vec{v}=\lambda\vec{v} ## ## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} -...
  42. P

    A matrix with Repeated eigenvalues and its corresponding eigenvectors.

    Homework Statement I am asked to find the diagonal matrix of eigenvalues, D, and the matrix of corresponding eigenvectors, P, of the following matrix: \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & -2\\ 0 & 0 & -1 \end{pmatrix} Homework Equations The Attempt at a Solution We just started this topic...
  43. N

    Trying to understand Eigenvectors

    Homework Statement Find the Eigenvalues of A= 4 0 1 -2 1 0 -2 0 1 Then find the eigenvectors corresponding to each of the eigenvalues. Homework Equations The Attempt at a Solution I found the Characteristic Polynomial of the matrix, computed the Eigenvalues which are...
  44. C

    Finding Eigenvectors and Eigenvalues

    Homework Statement The Matrix A is as follows A= [4 -4 0 2 -2 0 -2 5 3] and has 3 distinct eigenvalues λ1<λ2<λ3 Let Vi be the unique eigenvector associated with λi with a 1 as its first nonzero component. Let D = [ λ1 0 0 0 λ2 0 0 0 λ3] and P=...
  45. D

    Calculate the eigenvectors of a specific matrix

    Hello, I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one Homework Statement A= 2 1 0 1 0 3 -1 0 0 1 1 0 0 -1 0 3 λ1=2 multiplicity 3 λ2=3 multiplicity 1...
  46. J

    What are Eigenvectors and Eigenvalues in Relation to Matrices?

    Given a vector ##\vec{r} = x \hat{x} + y \hat{y}## is possbile to write it as ##\vec{r} = r \hat{r}## being ##r = \sqrt{x^2+y^2}## and ##\hat{r} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}##. Speaking about matrices now, the the eigenvalues are like the modulus of a vector and the eigenvectors...
  47. A

    Fortran Finding Eigenvalues & Eigenvectors with Fortran99 for Sparse Matrices

    Hi everybody.. How can i use fortran99 to find the eigenvalues & eigenvectors of sparse matrices? Thanx :)
  48. jk22

    What happens with non orthogonal eigenvectors

    I considered the covariance of 2 spin 1/2 as a non linear operator : A\otimes B-A|\Psi\rangle\langle\Psi|B. The eigenvectors are but non orthogonal and I wondered what happens in that case with the probabilities : from Born"s rule it comes that the transition probability from one vector to the...
  49. H

    Why Can't I Find All Eigenvectors for My 3x3 Matrix?

    Hey everyone. I have a matrix A = {{7,-5,0},{-5,7,0},{0,0,-6}} I have found the Eigenvalues, 2,12,-6 but I'm only getting one Eigenvector, (0,0,1).. I know there is 2 others (-1,1,0) and (1,1,0) but I am unable to get them by hand. Once I get the matrix in the form (A-λI)*v = o, I just get...
  50. D

    First-variational and second-variational eigenvectors

    Dear all, Recenty,I am reading the source code of the first-principle software.I meet some words that I haven't found in those DFT books.For example,it mentions the first-variational and second-variational eigenvectors. Similarly,the first-variational and second-variational eigenvalues are...
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