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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Finding a basis of eigenvectors

    Homework Statement A = \left( \begin{array}{ccc} 2 & 0 & -1 \\ 4 & 1 & -4 \\ 2 & 0 & -1 \end{array} \right) Find the eigenvalues and corresponding eigenvectors that form a basis over R3 Homework Equations The Attempt at a Solution OK so I've found the characteristic...
  2. T

    Negative determinants when calculating eigenvectors?

    Let M be a transformation matrix. C is the matrix which diagonalizes M. I'm trying to use the formula D = C-1MC. I noticed that depending on how I arrange my vectors in C, I can change the sign of the determinant. If I calculate D using a configuration of C that gives me a negative value for...
  3. T

    Textbook to help me understand eigenvectors and diagonalization

    Hi, I'm currently self-teaching myself some mathematics needed to study physics. I'm working through the book Mathematical Methods in the Physical Sciences by Mary L Boas. The book is a well known one, and it's used in many physics programs to teach their math courses. However, I've read the...
  4. T

    Finding eigenvectors of [[1,-1,-1],[-1,1,-1],[-1,-1,1]]

    he eigenvalues of the 3x3 matrix [[1,-1,-1],[-1,1,-1],[-1,-1,1]] are 2,2, and -1. how can i compute the eigenvectors? for the case lambda=2, for example, i end up with an augmented matrix [[-1,-1,-1,0],[-1,-1,-1,0],[-1,-1,-1,0]] so I'm stuck at this point. much appreciated.
  5. R

    Proof of generalised eigenvectors

    Hi this is my first time posting on here so hopefully I get it right. Given the linear system x'(t) = Ax(t)' with an eigenvalue (lambda) of algebraic multiplicity 2 and geometric multiplicity 1 (repeated root), one solution is w.exp(lambda t) and the other w.t.exp(lambda t) + u.exp(lambda t)...
  6. S

    Using power method to calculate dominant eigenvalue and eigenvectors

    Homework Statement Use the power method to calculate the dominant eigenvalue and its corresponding eigenvectors for the matrices. The questions are attached with this thread. I have attempted both and seem to have done the first question correctly. I am attempting the second question and am...
  7. E

    Lower triangular matrix eigenvectors problem

    Ok, this is starting to come back to me, but I'm stuck again Homework Statement M=\begin{bmatrix} (1-\frac{4}{3}) & 0 \\ -\frac{c}{3} & -c \\ \end{bmatrix} Find eigenvectors and eigenvalues. Homework Equations The Attempt at a Solution Eigenvalues are λ_1=...
  8. E

    Finding the eigenvectors in triangular matrices

    I thought I would ask this in the homework section. Homework Statement I should be able to write down the eigenvectors and eigenvalues of diagonal and triangular matrices on sight. M = \begin{bmatrix} 1 &0 \\[0.3em] 0 & x \\[0.3em]...
  9. E

    Another reminder on finding eigenvectors

    Another question with respect to finding eigenvectors. If I remember correctly, I should be able to look at certain 2 by 2 matrices and practically write down the eigenvalues and eigenvectors. For example, I have a diagonal matrix, I know immediately what the eigenvalues and eigenvectors...
  10. E

    Reminder on how to find eigenvectors

    Ok everybody, it's been awhile since I've taken linear algebra. I need some help dusting off the cobwebs. (I'm trying to follow this in a paper; this isn't a homework question, but I'll be glad to move it...) Let's say I have a matrix M = \begin{bmatrix} -σ & σ & 0...
  11. S

    Determining Matrix Powers without Eigenvectors ? (Worked out but inelegant soln)

    http://dl.dropbox.com/u/33103477/Untitled.png My solution: M=\begin{bmatrix} t+15 & -12 \\ 24 & t-19 \end{bmatrix} The eigen values are 1,3. Hence as the matrix has real and distinct eigenvalues it is diagonalisable. Now the characteristic equation is t^2 - 4t +3 =0...
  12. S

    How to calculate powers of a 2x2 matrix WITHOUT eigenvectors ?

    How do I determine powers of matrices(2x2) without calculating their eigenvectors and doing the pdp^-1 thing ? Obviously multiplying over and over is not a solution.
  13. T

    Jordan Normal Form & Generalized Eigenvectors

    I've been having some trouble with conceptually understanding the idea of a generalized eigenvector. If we have a linear operator A and want to diagonalize we get it's eigenvalues and eigenvectors but if the algebraic multiplicity of one of the eigenvalues is greater than the geometric...
  14. J

    Calculating eigenvalues and eigenvectors

    Homework Statement I'm having a problem with a question. I need to find the transition matrix in the form T=UAU^-1 where U=[V1 V2] Homework Equations T=UAU^-1 where U=[V1 V2] The Attempt at a Solution my original transition matrix is [0.9 0.002; 0.1 0.998] from that i calculated...
  15. B

    Which Eigenvector is Wrong for Finding Eigenvalues of a Matrix?

    Too many Eigenvectors!? Homework Statement I have to find the eigenvalues and eigenvectors of: -1 2 -2 1 2 1 -1 -1 0 and I can find four eigenvectors I'm not sure how to tell which of my eigenvectors is wrong as they all seem to satisfy Av=λv (i also checked that they arent...
  16. H

    Meaning of eigenvectors and values of a 2x2 matrix (2nd order tensor)

    Hi! I am a new user who is not an expert with Linear Algebra at all. I have some questions about eigen values/vectors and their meaning with relation to a 2x2 matrix, or tensor, which was obtained by the tensor product of 2 vectors. First, I have two 2-dimensional 2x1 vectors "v1" and "v2"...
  17. H

    Can Eigenvectors of the Same Eigenvalue Be Orthogonal in a 2x2 Matrix?

    This seems a simple question but I can't find the solution by myself. Please help. Say we have a 2 by 2 matrix with different eigenvalues. Corresponding to each eigenvalue, there are a number of eigenvectors. Q1. Could the eigenvectors corresponding to the same eigenvalue have different...
  18. J

    Express Eigenvectors in terms of Eigenkets.

    Homework Statement One of the problems in our test is this. Express the eigenvectors of J_y in terms of the eigenkets of J^2 and J_z . Homework Equations The Attempt at a Solution I know the matrix of J_y and the operators or eigenkets for J_y , J^2 and J_z . I just don't seem...
  19. T

    Eigenvalue of Sum of Eigenvectors

    Homework Statement The Attempt at a Solution So, first I wrote, T(X) = λ_1 X, T(Y) = λ_2 Y If λ_1 = λ_2: T(X+Y) = T(X) + T(Y) = λ_1 X + λ_2 Y = λ_1 (X+Y), so this does indeed seem to be an eigenvector. But I'm less convinced for the case λ_1 ≠ λ_2. Again, I get the...
  20. S

    Eigenvectors, spinors, states, values

    For spin-1/2, the eigenvalues of S_x, S_y and S_z are always \pm \frac{\hbar}{2} for spin-up and spin-down, correct? What is the difference between eigenvectors, eigenstates and eigenspinors? I believe eigenstates = eigenspinors and eigenvectors are something else? I'm just getting confused...
  21. S

    What is the Correct Way to Find Eigenvalues and Eigenvectors of a Matrix?

    Find the eigenvalues and corresponding eigenvector of the matrix. A= [-4 4 8 ] [0 0 -10] [0 0 2 ] [1 -1 0] ~ [0 0 1 ] [0 0 0 ] I calculated by A = -\lambdaI So, [1-lamda -1 0 ] [0 -lamda 1] [0 0 -lamda] so, lamda = 0,0, and 1 So I got...
  22. N

    Find the equilibrium solution and eigenvalues and eigenvectors of system?

    Hey guys, I need to find the equilibrium solution (critical point) for the given system. Also I need to take the homogeneous equation x' = Ax (matrix notation) and find the eigenvalues and eigenvectors. system: x' = -x - 4y - 4 y' = x - y - 6 Can you help? Thanks
  23. Y

    MHB Which statement about eigenvalues and eigenvectors is not true?

    One more question please... which one of these statements is NOT true (only one can be false): a. a square matrix nXn with n different eigenvalues can become diagonal. b. A matrix that can be diagonal is irreversible. c. Eigenvectors that correspond to different eigenvalues are linearly...
  24. T

    Eigenvalues, eigenvectors, and eigenspaces

    Homework Statement The Attempt at a Solution T(1,0,0) = (3,-1,0) T(0,1,0) = (0,1,0) T(0,0,1) = (-1,2,4) Thus, we have the matrix, \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right| Δ_T (t) = det( \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right|...
  25. matqkks

    Exploring the Power of Eigenvalues and Eigenvectors in Matrix Analysis

    We are aware that by knowing the eigenvalues and eigenvectors we can evaluate the determinant, say if it is invertible and diagonalize to find powers of matrices. Is there a list of properites of a matrix we can find by eigenvalues and eigenvectors? Are there things that e.values and e.vectors...
  26. A

    Can we still diagnolize a matrix if its eigenvectors matrix is singular?

    well, if a matrix has n linearly independent eigen-vectors then it's easy, what if a matrix is not diagnolizable in that way? Can we still diagnolize it by other means? And what if a matrix is not diagonalizable at all? Are there still ways to find its exponential matrix?
  27. S

    Please help with this matrix question Solve for Eigenvales, and Eigenvectors

    M = (a c) (c b) Sorry for the double sets of brackets, its all in one. I'll also show as far as i got below: [a-λ c] => (a-λ)(b-λ) - c^2 = λ^2 + (-a-b)λ + (ab-c^2) =0 [c b-λ] => then using the quadratic formula: λ = [-(-a-b) +/- Sqrt{(-a-b)^2 - 4(1)(ab-c^2)}]/ 2 then...
  28. G

    Eigenvectors of a 2D hermitian operator (general form)

    Homework Statement Calculate the eigenvectors and eigenvalues of the two-dimensional matrix representation of the Hermitean operator \hat{O} given by |v_k'>\left(O|v_k>= {{O_11,O_12},{O_21,O_22}} where all Oij are real. What does Hermiticity imply for the o - diagonal elements O12...
  29. Y

    MHB Eigenvalues and Eigenvectors of a Non-Diagonalizable Matrix

    Hello, sorry that I am asking too many questions, I am preparing for an exam... I have a matrix, 0 1 0 0 0 0 0 0 1 and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P) I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the...
  30. W

    Can an eigenvalue have multiple eigenvectors?

    I am little confused about the choice of eigenvectors chosen by my book. I am wondering if an eigenvalue can have multiple eigenvectors and if all are equally correct. Case in point the example below: Homework Statement find a fundamental matrix for the system x'(t) = Ax(t) for the given...
  31. D

    Eigenvalues and eigenvectors of J.n

    Homework Statement Calculate the eigenvalues and eigenvectors of the operator, J.n, where n is a unit vector characterized by the polar angles theta and phi, and J is the spin-1 angular momentum operator. Homework Equations Matrix representations for J^2 and J(z) The Attempt at a...
  32. F

    Eigenvalues and eigenvectors of observables

    Homework Statement Calculate the Eigenvalues and eigenvectors of H= 1/2 h Ω ( ]0><1[ + ]1><0[ ) Homework Equations I know H]λ> = λ]λ> The Attempt at a Solution I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?
  33. C

    What does eigenvalues and eigenvectors mean?

    I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how. What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways? I...
  34. L

    Completeness of eigenvectors in a complete, commuting set

    "Completeness" of eigenvectors in a complete, commuting set Hi guys, I asked this question on Physics Stack Exchange many days ago and even after substantial discussions and revisions, this has remain unanswered. This is...
  35. H

    How do I numerically find eigenvectors for given eigenvalues?

    My aim was to numerically calculate eigenvalues and eigenvectors for a square A matrix. I managed to find the eigenvalues by using QR algorithm. Now, I can find all of the eigenvalues for any given square matrix. But, for the next step, how do I find the corresponding eigenvectors? Is there...
  36. C

    Proving simultaneous eigenvectors for commuting operators

    Homework Statement In my quantum class we learned that if two operators commute, we can always find a set of simultaneous eigenvectors for both operators. I'm having trouble proving this for the case of degenerate eigenvalues.Homework Equations Commutator: [A,B]=AB-BA Eigenvalue equation:A...
  37. T

    Eigenvectors of a matrix in Jordan Normal Form

    Let A =\begin{bmatrix} -14 & 5 & -2 & 1 \\ -38 & 13 & -4 & 5 \\ 25 & -10 & 7 & 5 \\ -17 & 5 & -2 & 4 \end{bmatrix} . The Jordan Normal Form of A is J =\begin{bmatrix} 2 & 1 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 1 \\ 0 & 0 & 0 & 3 \end{bmatrix} . Now, I've been told that eigenvectors...
  38. K

    How Do I Solve for Eigenvectors After Finding Eigenvalues?

    In the last 2 weeks we've begun learning about eigenvalues/vectors. It will come up in my exam in January so I'm trying hard to get my head around this. I've tried various different sources to learn this but I'm hoping someone here can offer a different view on it. Basically, I can work out the...
  39. D

    How to find eigenvalues and eigenvectors for 5x5 matrix

    I got a 5x5 matrix, if use characteristic equation to find the eigenvalues and eigenvectors are very tedious and trouble, so got any method which are easy to calculate?
  40. T

    Inverse Power Method and Eigenvectors

    Homework Statement The Markov matrix A = [.9 .3; .1 .7] has eigenvalues 1 and .6, and the power method uk=Aku0 converges to [.75 .25]T. Find the eigenvectors of A-1. What does the inverse power method u-k=A-1u0 converge to (after you multiply by .6k)? Homework Equations The...
  41. T

    Second-Order Equations and Eigenvectors

    Homework Statement Convert y"=0 to a first-order system du/dt=Au d/dt [y y']T = [y' 0]T = [0 1; 0 0] [y y']T This 2x2 matrix A has only one eigenvector and cannot be diagonalized. Compute eAt from the series I+At+... and write the solution eAtu(0) starting from y(0)=3, y'(0)=4. Check...
  42. Geofleur

    What is the nullspace of (A-2E)^2?

    Please note: Below, I keep trying to put [ capital B ] but it gets turned into [b]! In Dennery and Krzywicki, they give an example of how to put a matrix in Jordan canonical form (pp. 167-170). They start with a 4x4 matrix [A] that looks kind of messy and transform it to a quasi-diagonal form...
  43. J

    Using eigenvalues and eigenvectors to solve system of ODEs

    Homework Statement Use eigenvalues and eigenvectors to find the general solution of the system of ODEs.. x1 = 3x1 - x2 x2 = -x1 + 2x2 - x3 x3 = -x2 + 3x3 Homework Equations The Attempt at a Solution I converted that into the matrix...
  44. S

    How do eigenvalues and eigenvectors relate to matrices?

    I have to be able to figure out eigenvalues and eigenvectors for 2x2 and 3x3 matrices for my physics course, but I have never taken linear algebra so I obviously have no idea what they even are. I need someone to basically teach me how to solve these problems because I have no knowledge of this...
  45. S

    Evaluating B4: Finding the Eigenvalues & Eigenvectors

    Homework Statement Let B be a matrix with characteristic polynomial λ2-λ√6+3. Evaluate B4. Homework Equations Bn=PDnP-1 The Attempt at a Solution I can find the eigenvalues from the characteristic equation and those would form the diagonal entries of D. But how would I find P, which contains...
  46. U

    Eigenvalues / eigenvectors concept explaination please

    Hello This is a concept question I do not understand. I'm just wondering why the answer is what it is. (the answer is written below the question, I just have no idea where it comes from)
  47. A

    Fortran Fortran90: Subroutine DSYEV and associating eigenvalues and eigenvectors.

    Greetings. I am using the LAPACK (Linear Algebra Package) software package to find the eigenvalues and eigenvectors of a large symmetrical real matrix. Specifically, I calculate a scalar from each eigenvector, and I want to graph it against its associated eigenvalue. I am using the subroutine...
  48. T

    Solving Complex Eigenvectors: Find Matrix A & Compute Solution

    Homework Statement The higher order equation y"+y=0 can be written as a unknown d/dt[y y']=[y' y"]=[y' -y] If this is du/dt=Au, what is the 2x2 matrix A? Find its eigenvectors and eigenvalues, and compute the solution THAT STARTS FROM y(0)=2, y'(0)=0. Homework Equations y'=Ay...
  49. C

    Finding eigenvectors for a given matrix

    Homework Statement Finding Eigenvectors of the given matrix Homework Equations The matrix is A=\begin{pmatrix} 5.956 & -1.284\\ -1.284&0.435 \end{pmatrix} The Attempt at a Solution I have found the eigenvalues to be \lambda _{1}=0.624001 and \lambda _{2}=0.150994...
  50. B

    If (Q^-1)AQ=D, then columns of Q are Eigenvectors of A

    Homework Statement Prove: if (Q^-1)AQ=D, then each column of Q is an eigenvector of A. Homework Equations A vector v is an eigenvector of A iff there exists a scalar λ such that: Av=λv The Attempt at a Solution Suppose (Q^-1)AQ=D. We need to show each column of Q is an...
Back
Top