# What is Eigenvector: Definition and 147 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. ### 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...
2. ### 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##...
3. ### A Generalized Jacobi Method

If I have two matrices A and B, how can I find an eigenvector for the two matrices?
4. ### 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 ##...
5. ### 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...
6. ### 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...
7. ### 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...
8. ### I Generalized eigenvector

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...
9. ### 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. ### 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...
12. ### 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.
13. ### 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...
14. ### 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
15. ### 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...
16. ### 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...
17. ### 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...
18. ### Eigenvector of raising operator

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:
19. ### 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...
20. ### 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...
21. ### Eigenvalues/eignevectors of Jones matrix

I did an exercice for an optic course and the question was to find which optical component, using eigenvalues and eigenvectors, the following Jones matrix was (the common phase is not considered) : 1 i i 1 I found that this is a quarter-wave plate oriented at 45 degree from the incident...
22. ### Eigenvectors for a 3x3 matrix

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. ### 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)...

25. ### 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. ### 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. ### B Elementary eigenvector question

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. ### 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. ### Problem about an Eigenvector

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. ### 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 [Broken] 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...
31. ### 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. ### 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. ### 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. ### 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...

37. ### 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. ### 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. ### 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. ### About basic eigenvector questions

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. ### 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. ### Proof of Eigenvector: A*v=λ*v

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. ### Eigenvalue and and eigenvector

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. ### Formula for unit eigenvector

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. ### Generalized eigenvector

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. ### 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. ### 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. ### 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. ### 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. ### 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?