Eigenvector Definition and 144 Threads

  1. durandal

    Engineering The Telegraph Equation

    I have tried row reduction to solve for the first eigenvector but I don't feel like I get any closer to the solution:
  2. Z

    Prove 9 is eigenvalue of ##T^2\iff## 3 or -3 eigenvalue of ##T##.

    Suppose ##9## is an eigenvalue of ##T^2##. Then ##T^2v=9v## for certain vectors in ##V##, namely the eigenvectors of eigenvalue ##9##. Then ##(T^2-9I)v=0## ##(T+3I)(T-3I)v=0## There seem to be different ways to go about continuing the reasoning here. My question will be about the...
  3. Z

    Operator T, ##T^2=I##, -1 not an eigenvalue of T, prove ##T=I##.

    Now, for ##v\in V##, ##(T+I)v=0\implies Tv=-v##. That is, the null space of ##T+I## is formed by eigenvectors of ##T## of eigenvalue ##-1##. By assumption, there are no such eigenvectors (since ##-1## is not an eigenvalue of ##T##). Hence, if ##(T-I)v \neq 0## then ##(T+I)(T-I)v\neq 0##...
  4. H

    A Finding Eigenvectors for Two Matrices using the Generalized Jacobi Method

    If I have two matrices A and B, how can I find an eigenvector for the two matrices?
  5. H

    How to find the eigenvector for a perturbated Hamiltonian?

    Hi, I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian. However, I don't see how to find the eigenvectors. To find the eigenvalues for the first excited state I build this matrix ##...
  6. H

    If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

    Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...
  7. U

    I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

    Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
  8. M

    I Graph Representation Learning: Question about eigenvector of Laplacian

    Hi, I was reading the following book about applying deep learning to graph networks: link. In chapter 2 (page 22), they introduce the graph Laplacian matrix ##L##: L = D - A where ##D## is the degree matrix (it is diagonal) and ##A## is the adjacency matrix. Question: What does an...
  9. Alwar

    I How to Find the Generalized Eigenvector in a Matrix ODE?

    Hi, I have a set of ODE's represented in matrix format as shown in the attached file. The matrix A has algebraic multiplicity equal to 3 and geometric multiplicity 2. I am trying to find the generalized eigenvector by algorithm (A-λI)w=v, where w is the generalized eigenvector and v is the...
  10. Passers_by

    Constructing an Eigenvector of S with Eigenvalue λ1

    There is a eigenvector n3 of S with eigenvalue equal to λ3 and a eigenvector n1 of S with eigenvalue equal to λ1. n1 and n3 are orthogonal to each other . Construct the vector v2 so that they're orthogonal to each other(n1,v2 and n3).We can prove that v2 is an eigenvector of S . But how do we...
  11. D

    I Normalization of an Eigenvector in a Matrix

  12. K

    Show that V is an internal direct sum of the eigenspaces

    I was in an earlier problem tasked to do the same but when V = ##M_{2,2}(\mathbb R)##. Then i represented each matrix in V as a vector ##(a_{11}, a_{12}, a_{21}, a_{22})## and the operation ##L(A)## could be represented as ##L(A) = (a_{11}, a_{21}, a_{12}, a_{22})##. This method doesn't really...
  13. M

    How to find one corresponding eigenvector?

    The determinant is 0, which means that A-4I has a nullspace, and there is an eigenvector with eigenvalue 4. In the textbook, the answer says "Yes, [1, 1, -1]" for this problem. But I don't know how to find the corresponding eigenvector for this problem. Below is my work.
  14. S

    Setting Free variables when finding eigenvectors

    upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
  15. J

    Find the eigenvector with zero eigenvalues at any time t from the Hamiltonian

    I have a question relates to a 3 levels system. I have the Hamiltonian given by: H= Acos^2 bt(|1><2|+|2><1|)+Asin^2 bt(|2><3|+|3><2|) I have been asked to find that H has an eigenvector with zero eigenvalues at any time t
  16. Mentz114

    I Choosing the eigenvector stochastically

    The equation ##\hat{O}|\psi\rangle \rightarrow \alpha_n|\mathbf{e}_n\rangle## where ##|\mathbf{e}_n\rangle## is an eigenvector of the operator and ##\alpha_n## is its eigenvalue, is central in the QT formalism. This is as much as we can get from quantum theory but an ideal instrument should...
  17. R

    Choosing Free Variable for Generalized Eigenvector

    As you can see from my eigenvalues, here I've got a repeated roots problem. I'm wondering if it matters which variable I can choose to be the free variable when I'm solving for the generalized eigenvector. I think both are equally valid but they look different from one another and I'd like to...
  18. Z

    Solving equations for Eigenvector: Vanishing

    Homework Statement I am trying to solve a Eigen vector matrix: ##\begin{bmatrix}9.2196& 6.488\\4.233& 2.9787\end{bmatrix}\cdot\begin{bmatrix}x\\y\end{bmatrix}-\lambda\begin{bmatrix}x\\y \end{bmatrix}=0## I have found ##\lambda_1 = 0## and ##\lambda_2 = 12.1983## However, I can't solve the...
  19. Pouyan

    Does the Raising Operator Have Right Eigenvectors?

    Homework Statement show that the raising operator at has no right eigenvectors Homework Equations We know at|n> = √(n+1)|n+1> The Attempt at a Solution we define a vector |Ψ> = ∑cn|n> (for n=0 to ∞) at|Ψ>=at∑cn|n>=∑cn(√n+1)|n+1> But further I give up!:cry:
  20. Z

    Evaluating a wrong value of Eigenvector

    Homework Statement I think I am evaluating a wrong value for eigen vector. Wrong in the sense that first row is giving different value and next row is giving a different value. The matrix is: [0 1] [-2 -3] My eigen values are λ=-1 & λ=-2. I get the fine vectors for λ=-1 but λ=-2 x...
  21. F

    Calculating eigenvectors/values from Hamiltonian

    Homework Statement I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point. For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube. I'm then constructing the Hamiltonian...
  22. M

    Calculating Eigenvectors for a 3x3 Matrix: Understanding the Process

    Hi, I am trying to find the eigenvectors for the following 3x3 matrix and are having trouble with it. The matrix is (I have a ; since I can't have a space between each column. Sorry): [20 ; -10 ; 0] [-10 ; 30 ; 0] [0 ; 0 ; 40] I’ve already...
  23. T

    Finding eigenvector QM 2x2 matrix

    Homework Statement I am having a issue with how my lecture has normalised the energy state in this question. I will post my working and I will print screen his solution to the given question below, we have the same answer but I am unsure to why he has used the ratio method. Q4. a), b), c)...
  24. ertagon2

    MHB Eigenvector Calculations and Verification for y1+y2=5

    Can you please confirm my answer? det(yI-A) = 0 (y-1)(y-4)=0 y1+y2=5
  25. 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...
  26. J

    Operator in three level system -- Eigenvalues/Eigenvectors

    There is an operator in a three-state system given by: 2 0 0 A_hat = 0 0 i 0 -i 0 a) Find the eigenvalues and Eigenvectors of the operator b) Find the Matrix elements of A_hat in the basis of the eigenvectors of B_hat c) Find the Matrix Elements of A_hat...
  27. J

    B Why Didn't the Eigenvector Approach Work for This Matrix?

    Hi If I have this matrix: \begin{array}{cc}0&1\\1&0\end{array} and I want to find its eigenvectors and eigenvalues, I can try it using the definition of an eigenvector which is: A x = λ x , where x are the eigenvectors But if I try this directly I fail to get the right answer, for example...
  28. A

    Find an appropriate matrix according to specific conditions

    I am facing some difficulties solving one of the questions we had in our previous exam. I am sorry for the bad translation , I hope this is clear. In each section, find all approppriate matrices 2x2 (if exists) , which implementing the given conditions: is an eigenvector of A with eigenvalue...
  29. arpon

    Do Eigenvalues of A and A^T have the same Eigenvectors?

    Homework Statement If ##AA^T=A^TA##, then prove that ##A## and ##A^T## have the same eigenvectors. Homework EquationsThe Attempt at a Solution ##Ax=\lambda x## ##A^TAx=\lambda A^Tx## ##AA^Tx=\lambda A^Tx## ##A(A^Tx)=\lambda (A^Tx)## So, ##A^Tx## is also an eigenvector of ##A##. What should be...
  30. 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...
  31. S

    I Some questions about eigenvector computation

    NOTE: For the answers to all these questions, I'd like an explanation (or a reference to a book or internet page) of how the answer has been derived. This question can be presumed to be for the general eigenproblem in which [ K ] & [ M ] are Hermitian matrices, with [ M ] also being positive...
  32. 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...
  33. B

    Normalising Imaginary Eigenvector

    Hello, whilst solving a system of coupled differential equations I came across an eigen vector of ##\vec{e_{1}} = (^{1}_{i})##. Assuming that this is a correct eigenvector, how do I normalise it? I want to say that ##\vec{e_{1}} = \frac{1}{\sqrt{2}} (^{1}_{i})## but if I sum ##1^{2} + i^{2}##...
  34. R

    Eigenvector of Pauli Matrix (z-component of Pauli matrix)

    I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0, I got...
  35. ognik

    MHB Help to understand this eigenvector case

    Did a practice problem finding eigenvalues &-vectors, ended with this row-reduced matrix: $ \begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$; Solving to get the eigenvectors, I could get $x_1 = x_2 = x_3$, in which case the eigenspace would be the zero vector. Instead the answer is $...
  36. ognik

    MHB Eigenvalues & Eigenvectors of $t: M_2 \implies M_2$ Matrix

    Q: Find the eigenvalues and eigenvectors of this map $t: M_2 \implies M_2$ $\begin{bmatrix}a&b\\c&d\end{bmatrix}$ $\implies\begin{bmatrix}2c&a+c\\b-2c&d\end{bmatrix}$ I don't know where to start, I suspect because I'm just not recognising what this represents, so if someone can tell me it is...
  37. K

    Physical significance of Eigenvalues and Eigenvector?

    I want to know what exactly Eigen value imply. What is its Physical significance ? Physical significance of eigen vector? Does eigen value concept apply in signal processing or evalvating frequency response off a system?
  38. rayne1

    MHB Eigenvector of 3x3 matrix with complex eigenvalues

    Matrix A: 0 -6 10 -2 12 -20 -1 6 -10 I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of: 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 So, how do I find the nonzero eigenvectors of the...
  39. kq6up

    Eigenvector Woes Homework Solution

    Homework Statement Find the eigenvectors of: ## \newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rrr} 5 & 0 & \sqrt{3} \\ 0 & 3 & 0 \\ \sqrt{3} & 0 & 3 \end{array}\right) ## Homework Equations ##(\mathbf{A}-\lambda\mathbf{I})\cdot\mathbf{x}=0## The Attempt at a Solution I get the...
  40. J

    How Do Eigenvectors Relate to Matrix Dimensions and Images?

    Homework Statement 1. If v is any nonzero vector in R^2, what is the dimension of the space V of all 2x2 matrices for which v is an eigenvector? 2. If v is an eigenvector of matrix A with associated eigenvalue 3, show that v is in the image of matrix A Homework Equations If v is an...
  41. W

    Problems about eigenvector in quantum mechanics

    I am learning about the basic quantum mechanics I know that an operator ,call it M^, is generally a matrix And we also can be represent it b a matrix representation M, associated with certain basis |e> M^ = sigma ( Mij |e> <e|) I,j Where Mij is matrix element of M So now I wonder...
  42. J

    Is Every Scalar Multiple of an Eigenvector Also an Eigenvector?

    Hello PF, brand new member here. A question about a proof: If A*v=λ*v, then w = c*v is also an eigenvector of A. This seems really simple to me, but perhaps I am doing it incorrectly: A*c*v=λ*c*v, divide both sides by c and you are left with your original eigenvector of A. Am I...
  43. G

    Why Are the Eigenvalues of This Matrix A and A + φ²B?

    Hi, I have a problem with the calculation of the eigenvalue of a matrix. That matrix is an N x N matrix which can be written as: ##M^{ab} = A\delta^{ab} + B \phi^a \phi^b## where ##\delta^{ab}## is the identity matrix and the ##\phi## is a column vector. The paper I'm studying says that...
  44. J

    Is There a Formula for Unit Eigenvectors?

    I never see before a formula for a eigenvector, however, given a generic matrix (http://www.wolframalpha.com/input/?i={{A%2C+B}%2C+{C%2C+D}}) the wolfram is able of find the eigenvectors... So, exist a formula for the unit eigenvectors?
  45. STEMucator

    Generalized Eigenvector Solutions for a System of Differential Equations

    Homework Statement Solve the system: ##x' = 5x - y## ##y' = 4x + y## Homework Equations ##t## is transpose. The Attempt at a Solution I'm a bit rusty with these and I had a small question. I put the system into the form ##x' = Ax## and proceeded to solve for the...
  46. N

    Eigenvalue, Eigenvector and Eigenspace

    Let's say my eigenvalue λ=-1 and we assume eigenvector of zero are non-eigenvector. An eigenspace is mathematically represented as Eλ = N(λ.In-A) which essentially states, in natural language, the eigenspace is the nullspace of a matrix. N(λ.In-A) is a matrix. Would it then be valid to say...
  47. M

    Description of eigenvector corresponding to each eigenvalue.

    I have a problem I need to solve. I can't find anything in my book that tells me how to do it. It might be worded differently in the book, but I'm not 100% sure how to solve this. Homework Statement Give a description of the eigenvectors corresponding to each eigenvalue. The Attempt at a...
  48. 9

    Solving Eigenvectors for Root -7 of 2x3 Matrix

    Homework Statement Find eigenvector for the root -7 of: |2 3| |3 -6| Homework Equations |2 3| |3 -6| The Attempt at a Solution I got 1 -3 But my books says -1 3 I am only wondering if this is possibly the same answer, because when I check my answer by multiplying...
  49. S

    Can anyone explain to me why eigenvector here is like this

    Im supposed to find the eigenvectors and eigenvalues of A I found that eigenvalues are 2 12 and -6 then I found eigen vectors substituting -6 to lambda and someone has told me I get 0 0 1 for eigenvector which I cannot understand why?? can anyone pleasezzzzzzzz explain why this is?
  50. 9

    Questing about eigenvector order

    In this video https://www.youtube.com/watch?feature=player_detailpage&v=INfPkT9EkhE#t=415, the presenter gets (1, -1) and (1, -8). Why exactly is it 1, -8 and not -8, 1, for example? How do you know what order to put it in?
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