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. R

    Help me prove something on eigenvectors?

    Homework Statement Prove that if the eigenvalues of a matrix A are \lambda_1 ... \lambda_n with corresponding eigenvectors x_1...x_n then \lambda^m_1...\lambda^m_n are eigenvalues of A^m with corresponding eigenvectors x_1...x_n Homework Equations Ax= \lambda x The Attempt at a...
  2. DeathbyGreen

    Mathematica Eigenvectors 4x4 Matrix in Mathematica

    Hi, I'm trying to calculate the eigenvectors of a 4x4 matrix, but I don't want the actual eigenvalues included in the solution, I simply want them listed as a variable. For example, I have the matrix: H_F = \left[ \begin{array}{cccc} \hbar\Omega&\hbar v_fk_- &0&0\\ \hbar...
  3. DeathbyGreen

    A Eigenvectors of a Floquet Hamiltonian

    I'm trying to recreate some results from a paper: https://arxiv.org/pdf/1406.1711.pdf Basically they take the Hamiltonian of graphene near the Dirac point (upon irradiation by a time periodic external field) and use Floquet formalism to rewrite it in an extended Hilbert space incorporating...
  4. L

    Eigenvectors of an operator

    Homework Statement ##H=\frac{J}{4}\sum_{i=1}^2 \sigma_i^x \sigma_{i+1}^x## Homework Equations ##\sigma^x ## is Pauli matrix and ##J## is number.[/B]The Attempt at a Solution For ##i=1## to ##3## what is dimension of eigen vector? I think it is ##8##. Because it is like that I have tri sites...
  5. Mr Davis 97

    Show that given n distinct eigenvalues, eigenvectors are independent

    Homework Statement Let ##T## be a linear operator on a vector space ##V##, and let ##\lambda_1,\lambda_2, \dots, \lambda_n## be distinct eigenvalues of ##T##. If ##v_1, v_2, \dots , v_n## are eigenvectors of ##T## such that ##\lambda_i## corresponds to ##v_i \ (1 \le i \le k)##, then ##\{ v_1...
  6. J

    MHB Eigenvalues and eigenvectors

    Can anyone explain to me where I would start with this type of question please https://uploads.tapatalk-cdn.com/20170308/d5986c078504823283e8884441e39c95.jpg https://uploads.tapatalk-cdn.com/20170308/26a8c5313c1e682ae35c5b47cd2d4973.jpg
  7. Adgorn

    I Regarding the linear dependence of eigenvectors

    Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue. However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
  8. 0

    Eigenvectors and orthogonal basis

    Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
  9. L

    A Matrices commute & Eigenvectors question

    Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?
  10. V

    Eigenvectors of Ly and associated energies

    Homework Statement Consider a particle with angular momentum l=1. Write down the matrix representation for the operators L_x,\,L_y,\,L_z,for this particle. Let the Hamiltonian of this particle be H = aL\cdot L-gL_z,\,g>0.Find its energy values and eigenstates. At time t=0,we make a measurement...
  11. Mr Davis 97

    Matrix A and its inverse have the same eigenvectors

    Homework Statement T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1 Homework EquationsThe Attempt at a Solution I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and...
  12. M

    Eigenvalue and Eigenvectors

    1. 1) Given 2x2 matrix A with A^t = A. How many linearly independent eigenvectors is A? 2) Is a square matrix with zero eigenvalue invertible? 2; When it comes to whether it is invertible; the det(A-λ* I) v = 0 where det (A-λ * I) v = 0 where λ = 0 We get Av = 0, where the eigenvector is...
  13. S

    I SO(3) rotation of eigenvectors

    Consider the eigenvectors ##(0, 1)## and ##(1, 0)## for the quantum system described the magnetic field ##\vec{B} = (0,0,B)##. Say I now rotate the magnetic field as ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##. Then the eigenvectors are supposed to change as...
  14. Konte

    I Hamiltonian matrix - Eigenvectors

    Hello everybody, From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix $$ \left( \begin{array}{cccccc} \langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\...
  15. S

    FEM - eigenvectors from eigensystem

    Homework Statement It's not really a homework so I will try to be as clear as possible. Hopefully, somebody will understand me and be able to help. I used Euler-Bernoulli theory to analyze the dynamics of a free-free beam (for the problem it is not important to understand what it is). If one...
  16. D

    I Eigenvectors for degenerate eigenvalues

    I am looking at some notes on Linear algebra written for maths students mainly to improve my Quantum Mechanics. I came across the following example - $$ \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} $$ The example then gives the eigenvalues as 0 and 1(doubly degenerate)...
  17. D

    I Eigenvalues, eigenvectors and the expansion theorem

    If i have an arbitrary ket then i know it can always be expressed as a linear combination of the basis kets.I now have an operator A which has 2 eigenvalues +1 and -1. The corresponding eigenvectors are | v >+ = k | b > + m | a > and | v >- = n | c > where | a > , | b > and | c > are...
  18. D

    I Spin Eigenvectors: | + > vs | - >?

    When considering the 2 eigenvectors of the Sz operator the | + > eigenvector points in the positive z direction and the | - > points in the negative z direction ; so is it correct to write | + > = - | - > ? And similarly for the eigenvectors of the Sx and Sy operators ? Thanks
  19. A

    I Why do eigenvectors stay the same when a matrix is squared?

    I am new to linear algebra but I have been trying to figure out this question. Everybody seems to take for granted that for matrix A which has eigenvectors x, A2 also has the same eigenvectors? I know that people are just operating on the equation Ax=λx, saying that A2x=A(Ax)=A(λx) and...
  20. Schwarzschild90

    The eigenvalues and eigenvectors of T

    Homework Statement Homework Equations The lattice laplacian is defined as \Delta^2 = \frac{T}{\tau} , where T is the transition matrix \left[ \begin{array}{cccc} -2 & 1 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \end{array} \right] and \tau is a time constant, which is...
  21. D

    I Eigenvalues and Eigenvectors question

    Let's say i have the 3x3 matrix a 0 0 b 0 0 1 2 1 it's eigenvalues are e1 =a, e2 = 0, e3 = 1. now if a = / = 0, 1 i have 3 distinct eigenvalues and the matrix is surely can be Diagonalizable. so if i try to solve for the eigenvector for the eigenvalue e1 =a: 0 0 0 b -a 0 1 2 1-a...
  22. K

    Eigenvalues and Eigenvectors: Finding the Roots of a Matrix

    Homework Statement we have this matrix 6 - 1 0 -1 -1 -1 0 -1 1 We need to find it's eigenvalues and eigenvectors Homework Equations The Attempt at a Solution[/B] I wrote the characteristic equation - det(A- λxunit matrix) to find the roots and got (-λ^3)+8(λ^2)+λ-6 instead of...
  23. odietrich

    I General form of symmetric 3x3 matrix with only 2 eigenvalues

    I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##. In general, such a matrix can be...
  24. SteliosVas

    System of Equations Homework: 4 Unknowns, 5 Equations

    Homework Statement Okay this is the problem it seems so easy but i just cannot for the life of me get it to click into my mind, I have 4 unknowns and 5 equations and i have to put it into a matrix and try solve it matricies or eigenvalues/eigenvectors. The 5 equations are: a= b/2 b=a/3 + d...
  25. K

    Calculators Getting Fractions on TI Nspire CX - Eigenvectors & Polynomial Roots

    Hey so I'm new to my TI nspire cx, still getting the hang of it. I've been trying to figure out how to get my eigenvector values to be fractions instead of decimals when I calculate them on here. Also, when I find the polynomial roots I get back decimals instead of fractions. I would like to...
  26. P

    I When should one eigenvector be split into two (same span)?

    This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf Given the operator \hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right) I find correctly that the eigenvalues are \lambda = b, \pm b. To find the eigenvectors for b, I do the following...
  27. Dusty912

    Finding eigenvectors for complex eigenvalues

    Homework Statement So I have been having trouble with finding the proper eigen vector for a complex eigen value for the matrix A=(-3 -5) . .....(3 1) had a little trouble with formating this matrix sorry The eigen values are -1+i√11 and -1-i√11 The Attempt at a Solution using AY-λY=0...
  28. D

    Eigenvectors and Eigenvalues: Finding Solutions for a Matrix

    Homework Statement Find the eigenvalues and associated eigenvector of the following matrix: Homework EquationsThe Attempt at a Solution We have a theorem in our lectures notes that states that if a matrix is invertible the only eigenvector in its kernel will be the zero vector. In order...
  29. H

    I A real matrix and its inverse share the same eigenvectors?

    Suppose ##v_i## is an eigenvector of ##A## with eigenvalue ##\lambda_i## and multiplicity ##1##. ##AA^{-1}v_i=A^{-1}Av_i=A^{-1}\lambda_iv_i=\lambda_iA^{-1}v_i## Thus ##A^{-1}v_i## is also an eigenvector of ##A## with the same eigenvalue ##\lambda_i##. Since the multiplicity of ##\lambda_i##...
  30. S

    I Normal Modes: Finding Eigenfrequencies

    If I have a system where the following is found to describe the motion of three particles: The normal modes of the system are given by the following eigenvectors: $$(1,0,-1), (1,1,1), (1,-2,1)$$ How can I find the corresponding eigenfrequencies? It should be simple... What am I missing?
  31. carllacan

    Eigenvectors of "squeezed" amplitude operator

    Homework Statement Prove that the states $$|z, \alpha \rangle = \hat S(z)\hat D(\alpha) | 0 \rangle $$ $$|\alpha, z \rangle = \hat D(\alpha) \hat S(z)| 0 \rangle $$ are eigenvectors of the squeezed amplitude operator $$ \hat b = \hat S(z) \hat a \hat S ^\dagger (z) = \mu \hat a + \nu \hat a...
  32. 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...
  33. 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...
  34. J

    How do I find eigenstates and eigenvalues from a spin operator?

    Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...
  35. G

    Sum of eigenvectors of linear transformation

    Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...
  36. Samuel Williams

    Eigenvalue and eigenvectors, bra-ket

    Question Consider the matrix $$ \left[ \matrix { 0&0&-1+i \\ 0&3&0 \\ -1-i&0&0 } \right] $$ (a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues? (b) Show that the eigenvectors |a1>, |a2>, |a3> form an...
  37. 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.
  38. 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...
  39. M

    Eigenvectors and Row/Column Vectors: What's the Connection?

    Is there a general between the eigenvectors of a matrix and the row (or column) vectors making up the matrix?
  40. S

    Eigenvalues and eigenvectors

    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 +...
  41. Y

    MHB Eigenvalues and eigenvectors

    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...
  42. TheSodesa

    Finding the eigenvectors of a matrix A

    Homework Statement A = \begin{bmatrix} 2 & 1 & 0\\ 0& -2 & 1\\ 0 & 0 & 1 \end{bmatrix} Homework EquationsThe Attempt at a Solution The spectrum of A is \sigma (A) = { \lambda _1, \lambda _2, \lambda _3 } = {2, -2, 1 } I was able to calculate vectors v_1 and v_3 correctly out of the...
  43. 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...
  44. kostoglotov

    Graphing Eigenvectors and Sine Curves: Understanding the Relationship

    In my text, it tells me to find the eigenvectors of a 2nd difference matrix and graph the eigenvectors to see how they fall onto sine curves. imgur link: http://i.imgur.com/oxbkTn6.jpg My question is simple but general. What does this even mean? How did they produce this graph from the...
  45. D

    Diagonal Scaling of a 2x2 Positive Definite Matrix

    Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by: $$ A = \begin{bmatrix} {A}_{11} & {A}_{12} \\ {A}_{12} & {A}_{22} \end{bmatrix} $$ And a Matrix ## B ## Given by: $$ B = \begin{bmatrix} \frac{1}{\sqrt{{A}_{11}}} & 0 \\ 0 & \frac{1}{\sqrt{{A}_{22}}}...
  46. kostoglotov

    Diff eqs with eigenvectors: double roots, but 2nd eigenvector?

    The problem is here, I'm trying to solve (b): imgur link: http://i.imgur.com/ifVm57o.jpg and the text solution is here: imgur link: http://i.imgur.com/qxPuMpu.pngI understand why there is a term in there with cte^t, it's because the A matrix has double roots for the eigenvalues. What I...
  47. kostoglotov

    Need a refresher: 1st order linear diff eq

    I thought I understood how to solve these sorts of equations, but apparently not.. 1. Homework Statement In Linear Algebra I'm solving diff eqs with eigenvectors to get all the combinations that will solve for a diff eq. The text then asked me to check my answer by going back and solving...
  48. J

    Extremely confused on finding eigenvectors

    Extremely confused on finding eigenvectors? Below I have a picture that gives the matrice and the eigenvectors. How did the solution find these eigenvectors?? i.e. the eigenvalues are 7 and -2 IMAGE LINKS http://tinypic.com/r/2liii68/9 http://tinypic.com/view.php?pic=2liii68&s=9#.VkY_YfmrSUk
  49. kostoglotov

    What is the solution for a system of ODEs with a matrix coefficient?

    I know how to solve \frac{d\vec{u}}{dt} = A\vec{u}, I was just watching a lecture, and the lecturer related that solving that equation is pretty much a direct analogy to \vec{u} = e^{At}\vec{u}(0), in so far as all we need to do after that is understand exactly what it means to take the...
  50. 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...
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