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. 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?
  2. 9

    Solving Eigenvector Order Homework

    Homework Statement A = 2 -2 -2 5 Eigenvalues are: 6, 1 Find eigenvectors. My only question is about order. My book lists them in the opposite order as I and I am not sure where I went wrong. Homework Equations A = 2 -2 -2 5 Eigenvalues are: 6, 1 The Attempt at a...
  3. Seydlitz

    Simple Proof for the existence of eigenvector

    Hello, My question is this. Is it possible to prove that there exist an eigenvectors for a symmetric matrix without discussing about what eigenvalues are and going into details with characteristic equations, determinants, and so on? This my short proof for that: (The only assumption is ##A##...
  4. 2

    Formulating an Eigenvector Equation

    Hello. I am working on a project with a double pendulum and I am currently looking for the normal mode frequencies. I don't think that's too important to answer my question, but in the derivation I hit a point that look like this:(K-M\omega^{2})\alpha=0. Here, K and M are 2x2 square matrices. I...
  5. B

    Eigenvector of complex Eigenvalues

    Homework Statement ##A=\begin{bmatrix} 16 &{-6}\\39 &{-14} \end{bmatrix}## Homework Equations The Attempt at a Solution I did ##A=\begin{bmatrix} 16-\lambda &{-6}\\39 &{-14-\lambda} \end{bmatrix}## and got that ##\lambda_1=1+3i## and ##\lambda_2=1-3i## The solution...
  6. K

    MHB Eigenvector and eigenvalue for differential operator

    My friends and I have been struggling with the following problem, and don't understand how to do it. We have gotten several different answers, but none of them make sense. Can you help us? **Problem statement:** Let $V$ be the vector space of real-coefficient polynomials of degree at most $3$...
  7. U

    Question about eigenvector and identity matrix

    Homework Statement I was doing this practice exam and I had to calculate the eigenvalues en vectors. The matrix had two eigenvalues, I calculated one eigenvector. But when I was performing row operations for the second eigenvector, the matrix with the second eigenvalue substitued became an...
  8. Petrus

    MHB Differential equation with eigenvector (complex number)

    Hello MHB, Solve the following system of linear differential equation f'=f-g g'=f+g with bounded limit f(0)=0, g(0)=1 could anyone check if My answer is correct? Just to make sure I understand correctly! ps we get \lambda=1-i and \lambda=1+i Regards, |\pi\rangle
  9. Petrus

    MHB Differential equation with eigenvector

    Hello MHB, solve this system of linear differential equation f'=f-g-h g'=-f+g-h h'=-f+g+h with boundary conditions f(0)=1, g(0)=2 and h(0)=0 we get that \lambda=1 and \lambda=0 now for eigenvector or we can call it basis for eigenvector \lambda=0 i get Is that correct? Regards, |\pi\rangle
  10. Sudharaka

    MHB Linear Transformation with No Eigenvector

    Hi everyone, :) This is one of those questions I encountered when trying to do a problem. I know that a eigenvector of a linear transformation should be non-zero by definition. So does that mean every linear transformation has eigenvectors? What if there's some linear transformation where no...
  11. A

    Find the Largest Eigenvalue & Eigenvector of A

    A=a.a', where a is an N by 1 vector,a'a=5,and T is transpose. a)Give the largest eigenvalue of A. b)what is the corresponding eigenvector? Please help me to solve the problem.
  12. Telemachus

    Quantum Mechanics: zero eigenvector

    Well, I know this have no sense. But I was trying to solve a problem on Cohen Tannoudji. The problem is in chapter IV, complement ##J_{IV}##, exercise 8. It says: Consider an electron of a linear triatomic molecule formed by three equidistant atoms. We use ##\left | {\phi_A} \right >, \left |...
  13. O

    Eigenvector proof from Dirac's QM

    Hi everyone, I'm currently working my way through Dirac's Quantum Mechanics, and I found this proof really irritating. We're trying to demonstrate that any eigenket can be expressed as a sum of eigenkets of a real linear function \xi which satisfies the equation \varphi(\xi) =...
  14. A

    Motivation behind eigenvalue and eigenvector

    An eigenvector is defined as a non-zero vector 'v' such that A.v = λ.v I don't understand the motive behind this. We are trying to find a vector that when multiplied by a given square matrix preserves the direction of the vector. Shouldn't the motive be the opposite i.e. finding the matrix...
  15. S

    Prove that v is an eigenvector of operator B

    Homework Statement Let B be the linear operator (1-x^{2}) \frac{d^2}{dx^2}-x\frac{d}{dx} Show that T_{4}(x) = 8x^{4} - 8x^{2} + 1 is an eigenvector of B, and find the corresponding eigenvalue. Attempt Righto, I find these rather difficult so a step by step solution would be nice but...
  16. E

    Zero as an element of an eigenvector

    Quick question on eigenvectors; Are there any general properties of a matrix that guarantee that a zero will or will not appear as an element in an eigenvector? Thank you!
  17. E

    Simple eigenvector question - please evaluate

    Hi, A - I =\begin{bmatrix} -0.5253 & 0.8593 & -0.1906 \\ -0.8612 & -0.5018 & 0.1010 \\ 0.1817 & 0.1161 & -0.0236\end{bmatrix} My eigenvector answer is t= k(−0.0137,0.225,1) My solution sheet's answer is t = k(-0.0088, 0.216, 1) Could I please ask that somebody checks this by...
  18. F

    Linear Algebra Eigenvector Properties

    Homework Statement True/False: If true give a proof, if false give a counterexample. a) If A and B have the same eigenvector X, then A+B should also have the same eigenvector, X. b) if A has an eigenvalue of 2, and B has an eigenvalue of 5, then 7 is an eigenvalue of A+B...
  19. Fernando Revilla

    MHB Yont's question at Yahoo Answers (eigenvector)

    Here is the question: Here is a link to the question: Find the eigenvector? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  20. E

    Finding an eigenvector of 3x3 matrix

    Hi, I'm trying to find an eigenvector of a matrix. I know that λ = 1, so my matrix (A - λI) is [-0.5253, 0.8593, -0.1906; -0.8612, -0.5018, 0.1010; 0.1817, 0.1161, -0.0236] And from rows 2 and 3 I get these simultaneous equations -0.8612t_{1}-0.5018t_{2}+0.1010t_{3}=0...
  21. A

    Eigenvector of a spin operator

    Homework Statement Homework Equations The Attempt at a Solution I don't know what's wrong with my work. I can't obtain the eigenvector provided in the model answer. My work Model Answer
  22. G

    Invariant spaces and eigenvector problem

    Homework Statement Let W be a 1-dimensional subspace of V that is A-invariant. Show that every non zero vector in W is a eigenvector of A. [A element of Mn(F)] The Attempt at a Solution We know W is A-invariant therefore for all w in W A.w is in W. W is one dimensional which implies to...
  23. M

    How do you find this eigenvector?

    [0 1] [-2 -2] This is the 2x2 matrix. [λ -1] [2 λ+2] This is the matrix that equals λI - A. Here are the eigenvalues I found: λ = -1 + i, -1 - i I am really confused at what to do next to find the eigenvectors. I keep looking online for help but I still can't figure it...
  24. F

    Finding Eigenvector for 3x3 Matrix: Step-by-Step Guide

    any1 can please tell me the eigen vector for following matrix: [0 0 a 0 0 0 0 0 0] please elaborate ur answer!
  25. J

    Making a eigenvector a linear combination of other eigenvectors

    Homework Statement Write the eigenvector of \sigmax with +1 eigenvalue as a linear combination of the eigenvectors of M. Homework Equations \sigmax = (0,1),(1,0) (these are the columns) The Attempt at a Solution ... Don't know what to do. Can someone show me how to do this using...
  26. R

    Showing that the normalized eigenvector for a distinct eigenvalue is unique

    Hey guys, I've been trying to brush up on my linear algebra and ran into this bit of confusion. I just went through a proof that an operator with distinct eigenvalues forms a basis of linearly independent eigenvectors. But the proof relied on a one to one mapping of eigenvalues to...
  27. Hercuflea

    I can't seem to find an eigenvector for this 2x2 matrix

    Homework Statement I'm doing an ODE for homework and I can't find the eigenvector for this matrix (sorry, I don't know how to make matrices on here. Consider these as one matrix.): [ 2 0] [v_{1}] = [0] [ 1 1] [v_{2}] = [0] Homework Equations The only way he has taught us...
  28. A

    On an eigenvector of matrix

    Homework Statement A=||A(i,j)|| (i,j=1,…,n) (n>2) is a binary matrix with zero diagonal and A(i,j)=1-A(j,i) for i≠j. W=(1,1,…,1)’ is an eigenvector for matrix B=A*A. Will W be an eigenvector for matrix A too? Why? 2. The attempt at a solution Let have a look at these two statements: "a"...
  29. A

    Understanding Eigenvectors: Troubleshooting and Verification

    Hello people! I am having a bit trouble with verifying my result when i compute the eigenvectors for the following matrix: A=[[3,4],[3,2]] I know for sure that the eigenvalues is respectively -1 and 6, so i start finding a solution for the following null spaces: 1)...
  30. S

    Difference between an eigenspace and an eigenvector ?

    So I'm a bit confused between these two and can't quite find any useful resources online. So is an eigenspace a special type of eigenvector cause that's how I understand it now.
  31. J

    Numerical approximation of the eigenvalues and the eigenvector

    Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...
  32. R

    What is the Eigenvector for a 2x2 Matrix with Eigenvalue -2?

    for the matrix {5,0} {2,-2} when determining the eigenvector for its 2nd eigenvalue, -2, you would start out by doing {5--2 ,0} {2 ,-2--2} giving {7,0} {2,0} In equation form this is 7u + 0v = 0 2u + 0v = 0 Ordinarily I would set u or v to a value and solve...
  33. T

    Finding eigenvalues and eigenvectors for a polynomial transformation

    Hi, So for some reason I have the hardest time trying to work with polynomials in linear algebra. I can't explain it, but whenever I see a question I draw a complete blank. Question: i) Find all the eigenvalues. ii) for each eigenvalue λ, find a basis of the eigenspace Eλ. T: P3(R) -->...
  34. Z

    Is this normalised eigenvector undefined?

    Homework Statement I've started off with a 2x2 matrix of (0) (i) (-i) (0) and I found the eigenvalues to be +1, -1 Then I found the resulting 2x1 eigenvectors to be (-i) (1) and (1) (-i) I now need the normalised eigenvectors. Homework Equations The...
  35. I

    Eigenvector eigenvalue proof problem

    Homework Statement Let A and B be symmetric matrices and X is a vector in the eigenvalue problem AX-λBX=0 a) Show that the eigenvectors are orthogonal relative to A and B. b) If the eigenvectors are orthonormal relative to B , determine C such that (C-λI)X=0, where C is a diagonal...
  36. S

    Eigenvector for Complex Eigenvalue help

    In my lecture notes my prof used the eigenvalue c= 1 + i and ended up with the matrix with (5 3+i) as row 1, and the second row is zeroes. After that, he simply wrote that the basis for this eigenvalue c is (3+i,-5) (in column form) without explaining. How did he get that basis? I tried working...
  37. A

    Eigenvector for A = [0 0; 1 -3] Not Working

    A = [0 0] [1 -3] (2 x 2 matrix, bad formatting) I need to find the eigenvector for lambda1 and lambda2. I figured out lambda1 = 0 and lambda2 = -3. For lambda1 the eigenvector works fine, but for lambda2 I get it as v = (0,0), which is not possible. Any ideas?
  38. Q

    By multiplying both sides by P.

    Homework Statement lets say i have a matrix A which is symmetric i diagonalize it , to P-1AP = D Question 1) am i right to say that the principal axis of D are no longer cartesian as per matrix A, but rather, they are now the basis made up of the eigen vectors of A? , which are the columns...
  39. H

    Please check my Eigenvector solutions.

    Homework Statement Find the characteristic equations, eigenvalues and eigenvector of the following matrix Homework Equations The Attempt at a Solution Somehow somewhere I think the solution is wrong, based on online Eigenvector calculator on the web. Please do provide me actual answers and...
  40. G

    Algebraic Muliplicity of an EigenVector

    1.Hello! I am having trouble understanding what A.M. is in the problem which asks, "Find the eigenvalues and eigenvectors associated with the matrix and find the a.m and g.m of each; for example... -1 0 0 1 0 1 - I * Lambda 0 2 1 The Attempt at a...
  41. D

    How can x1 be free if its coefficient is zero?

    Homework Statement Find the Eigenvector of the matrix.Homework Equations Ax=λx A= [2 0 1] [0 3 4] [0 0 1]The Attempt at a Solution Ok I'm just having a major brain fart here, been doing this all day. For λ=2, I solved for x and get this solution, [0 1 0][x1] [0] [0 0 1][x2]=[0] [0 0...
  42. P

    Calculating Eigenvectors and Eigenvalues for a Given Matrix

    Homework Statement Find the eigenvalues and the eigenvectors for the given matrix.Homework Equations \[ A = \left[ {\begin{array}{ccc} -1 & 6 & 2 \\ 0 & 5 & -6 \\ 1 & 0 & -2 \\ \end{array} } \right] \]The Attempt at a Solution I solved A-\lambda I = 0 and got eigenvalues of -4 and 3...
  43. C

    Solns to complex eigenvector eq's

    I have the following complex eigenvector: -1+2i -5 0 1 1+2i 0 What is the best way to go about solving these problems? I've done a few by inspection/trial&error, but I believe there has to be a more formal way to do it.
  44. G

    Eigenvector orthogonality and unitary operator diagonalization

    Homework Statement For reference: Problem 1.8.5 parts (3) , R. Shankar, Principles of Quantum Mechanics. Given array \Omega , compute the eigenvalues ( e^i^\theta and e^-^i^\theta ). Then (3) compute the eigenvectors and show that they are orthogonal. Homework Equations Eulers...
  45. T

    Determine unit normal (eigenvalue, eigenvector)

    Homework Statement For a material the stress is defined by the means of the stress matrix O O = (6 1 -2 1 2 2 -2 2 5) Expressed in MPA It can be derived that the principe stress are: O1= 4-sqrt(13), O2= 5 and O3=4+sqrt(13) I know you can derive the principal...
  46. M

    Know Eigenvalue and Eigenvector, How Do I Figure Out a Possible Original Matrix?

    Homework Statement This is a general question... I can easily go from a matrix A to its eigenvalues and then eigenvectors but how would I go from the eigenvalues and eigenvectors to a feasible original matrix? Any thoughts appreciated!
  47. C

    The VERY last part of finding an eigenvector

    I can do everything, until I get to the point of actually putting the system of equations into that eigenvector "form". I won't use my actual numbers, but say I have solved everything and got: ax = by ax = by Where a and b are 2 numbers, but a doesn't equal b. Does this mean that for everyone...
  48. P

    Solving Eigenvalue / Eigenvector Problem

    Homework Statement The problem is from a text on FEA, but I've "solved" the problem down to an eignenvalue/eigenvector problem. The point is to show that L_n = (2n-1)pi / (2a) and that the solution u(r,T) = sum [ a_n r^(L_n) ( cos (L_n * T) + (-1)^n sin (L_n * T) ] for n = 1 to infinity. L...
  49. B

    Diagonalize Matrix, Given an Eigenvalue and Eigenvector

    Homework Statement \begin{bmatrix} -7 && -16 && 4\\ 6 && 13 && -2\\ 12 && 16 && 1 \end{bmatrix} Diagonalize the matrix (if possible), given that one eigenvalue is 5, and that one eigenvector is {-2, 1, 2}Homework Equations A=PDP^{-1} The Attempt at a Solution If I were allowed to simply...
  50. W

    Proving Existence of Positive Eigenvector in 2x2 Matrix with Positive Elements

    Homework Statement Let A = matrix [a b] row 1 [c d] row 2 (2x2 matrix) with a>0, b>0,c>0,d>0. Show that A has an eigenvector [x,y] (2x1) with x>0, y>0 Homework Equations The Attempt at a Solution I've tried finding the characteristic polynomial by using det((lambda)I - A)...
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