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. C

    Eigenvectors of symmetric matrices

    Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?
  2. N

    Is Ax=wKx considered an eigenvalue problem in advanced linear algebra?

    From my Linear Algebra course I learned tha and eigenvalue w is an eigenvalue if it is a sollution to the system: Ax=wx, where A= square matrix, w= eigenvalue, x= eigenvector. We solved the system by setting det(A-I*w)=0, I=identity matrix Now in an advanced course I have come upon the...
  3. A

    Finding eigenvalues and eigenvectors 2x2 matrix

    Find the eigenvalues and corresponding eigenvectors of the following matrix. 1,1 1,1 Here is my attempt to find eigenvalues: 1-lambda 1 1 1-lambda Giving me: (Lambda)^2 -2(lambda) = 0 lambda = 0 lambda = 2 Is this correct??
  4. Telemachus

    Eigenvalues and eigenvectors [Linear Algebra]

    Homework Statement Hi there. I must give the eigenvalues and the eigenvectors for the matrix transformation of the orthogonal projection over the plane XY on R^3 So, at first I thought it should be the eigenvalue 1, and the eigenvectors (1,0,0) and (0,1,0), because they don't change. But I...
  5. H

    Eigenvectors of a 2 x 2 matrix

    I am evaluating the following 2 x 2 matrix: |2 0 | |0 3 | with eigenvalues 2 and 3. If I use 2 and calculate the eigenvector: R - λI = |2-λ 0 | |0 3-λ | R - λI = |0 0 | |0 1 | |0 0 ||a| = |0 1 ||b| |0| |0| a = 0 and b = 1 So...
  6. M

    Find Eigenvectors for x=2 of Matrix A: Help!

    Homework Statement Given the matrix 1 1 1 -1 3 1 -1 1 3 x=3 is an eigenvalue and (1,1,1) is a corresponding eigenvector x=2 is an eigenvalue of A of multiplicity 2 Find the eigenvector(s) corresponding to x=2 The Attempt at a Solution (A-AI)= -1 1 1 -1 1 1 -1 1 1...
  7. P

    Matrix Differential Equation with Generalized Eigenvectors

    Hey guys, need some quick help before an exam I have a differential eqn. x' = | 0 1 | *x , and initial conditions x(0) = |2| | -25 10 | |3| I find that there are two eigenvalues 5, and 5 The corresponding eigenvector to 5 is [1 5]...
  8. W

    Linear Algebra (eigenvectors, eigenvalues, and orthogonal projections)

    Homework Statement I am part way done with this problem... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now. W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it. a)...
  9. M

    Is there more than one possibility for eigenvectors of a single eigenvalue

    I guess this is best explained with an example. The matrix (0 -1) has the eigenvalues ------------------------------------------------------------------ (1 0) i and -i. For -i we obtain ix1-x2=0 and x1+ix2=0. I got a corresponding eigen vector (1 i), but when I controlled this result with...
  10. R

    Solving First-Order Linear Differential Equations with Eigenvectors

    Homework Statement solve the system of first-order linear differential equations: (y1)' = (y1) - 4(y2) (y2)' = 2(y2) using the equation: (λI -A)x = 0 Homework Equations using eigenvectors and eigenvalues in the book 'Elementary Linear Algebra' by Larson and Falvo - Section 7.4 #19...
  11. L

    Finding A Matrix, given eigenvalues, and eigenvectors

    Find a matrix that has eigenvalues 0,18,-18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1). ... I know the diagonlize rule, and the the rule to find a a power of A A= PDP^-1 D=P^-1AP ... but i am lost as to how to contine... help please?
  12. E

    Eigenanalysis: Finding Eigenvectors

    Homework Statement This is a simple example from the book, but it gets the point across nicely. In this problem eigenanalysis is used as a method to solve linear systems. The matrix... [4 2] [3 -1] Eigenvalues are -2, 5. Homework Equations (A-\lambda I)v=0 x'=[above matrix]x The...
  13. D

    Scalar Product of Momentum Eigenvectors in terms of Little Group Representation

    I'm trying to derive the equation for the scalar product of one particle momentum eigenvectors \Psi_{p,\sigma} ( p is the momentum eigenvalue and \sigma represents all other degrees of freedom), in terms of the little group of the Lorentz group with elements W that take the standard four...
  14. E

    Simple Yes or No Will Do Eigenvectors

    Simple Yes or No Will Do... Eigenvectors Homework Statement I think my prof made a mistake when writing this problem: Find a basis of the space V of all 3 x 3 matrices A for which the vectors <1, 1> and <1, 2> are eigenvectors and thus determine the dimensions of V. Is this problem...
  15. I

    Finding Eigenvectors for a Real Canonical Form of Matrix A

    Homework Statement Let A= [0 2 1;-2 3 0;1 0 2] Determine a real canonical form of A and give a change of basis matrix P that brings the matrix into this form. Homework Equations The Attempt at a Solution I found my eigenvalues to be 0, 2+i and 2-i. So, taking 2+i, I get the real...
  16. F

    Do Commuting Operators Always Share a Common Basis of Eigenvectors?

    Hey guys, I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra: 1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more...
  17. L

    Are the eigenvectors of A and A^T related?

    I have an (unknown) matrix A and with real non-negative values. I know its largest eigenvalue \lambda and the associated eigenvector, v. (I know nothing about the other eigenvectors). Does this give me any information about the eigenvector of AT associated with \lambda or is it completely...
  18. H

    Prove the eigenvectors are linearly independent

    Homework Statement Suppose that a matrix A has real entries (which we always assume) and a complex (non-real) eigenvalue  \lambda= a + ib , with b not equal to 0. Let W = U + iV be the corresponding complex eigenvector, having real and imaginary parts U and V , respectively. Show that U...
  19. H

    Eigenvectors and the dot product

    Homework Statement Suppose the the matrix A is symmetric, meaning that A = a b b d Show that for any symmetric matrix A there are always real eigenvalues. Also, show that the eigenvectors corresponding to two di erent eigenvalues are always orthogonal; that is, if V1 and V2 are the...
  20. G

    Did I make a mistake in finding the third generalized eigenvector for A?

    I remember reading a theorem that said that for an n x n matrix A, there exists a basis of Cn consisting of generalized eigenvectors of A. For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's)...
  21. A

    What are the eigenvectors for the given matrix A = [1 0 0; -2 1 3; 1 1 -1]?

    Homework Statement Given the matrix A = [1 0 0 -2 1 3 1 1 -1] Find an invertable matrix X and a diagonal matrix D such that A = XDX^-1 Homework Equations A = XDX^-1The Attempt at a Solution I've found that the eigenvalues are -2, 2...
  22. L

    Eigenvalues and eigenvectors of a matrix product

    We have two nxn matrices with non-negative elements, A and B. We know the eigenvalues and eigenvectors of A and B. Can we use this information to say anything about the eigenvalues or eigenvectors of C=A*B? The largest eigenvalue of C and the associated eigenvector are of particular interest...
  23. Y

    Trying to find the eigenvectors of a hamiltonian operator

    Homework Statement I am given the Hamiltonia operator of a system in two-dimensional Hilbert space: H = i\Delta(|w1><w2| + |w2><w1|) and am asked to find the corresponding eigenstates. I wrote this operator as a matrix, where H11 = 0, H22 = 0, and H12= i\Delta and H21= -i\Delta...
  24. L

    How do I find the eigenvalues and eigenvectors of a given matrix?

    Homework Statement Find the eigenvalues and eigenvectors of this matrix. [4 0 1/2 ] [0 -5 0 ] [1/2 0 1 ] [b]3. The Attempt at a Solution I have found the eigenvalues = -5, 5/2 + sqrt(5/2), 5/2 - sqrt(5/2) I know to get the eigenvectors you subtract the...
  25. D

    Question involving eigenvectors

    Homework Statement So I have to find the eigenvalues and eigenvectors of A=\left(\begin{array}{cc}0&1\\1&0\end{array}\right) which is not that special and hard. I solve the characteristic equation and find the eigenvalues: \lambda_{1,2}=\pm 1. So finding the eigenvectors is relatively...
  26. O

    Optimizing Eigenvector Computations for Large Matrices

    * corresponds to matrix product I'm working on a method of visualising graphs, and that method uses eigenvector computations. For a certain square matrix K (the entries of which are result of C_transpose*C, therefore K is symmetric) I have to compute the eigenvectors. Since C is mXn, where...
  27. P

    Understanding Eigenvectors and Eigenvalues in Linear Transformations

    Can someone explain these to me?
  28. V

    What is the significance of diagonalizing the matrix in an LC-circuit?

    I read that n different eigenvalue matrix has always n eigenvectors. But I cannot find any. Here is the state transition function, A: \left[\begin{array}{cc}\dot{I}\\ \dot{U}\end{array}\right] = \left[\begin{array}{cc}0&-1/L\\ 1/C&0\end{array}\right] \left[\begin{array}{cc}{I}\\...
  29. jinksys

    Find the Eigenvectors and eigenvalues of this matrix

    I'm trying to find the Eigenvectors and eigenvalues of this matrix: [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 ] I get 0, 1, and -1 as my eigenvalues. Starting with 0, I solve for reduced row echelon form and get the matrix: [ 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] My question is, and maybe my...
  30. G

    What is the method for finding the eigenvector for A-0I?

    Homework Statement A = [2,1,2;2,1,2;2,1,2] Find the Eigenvectors of A Homework Equations The Attempt at a Solution First I found the eigenvalues of A det(A - \lambda I) = 0 \lambda = 0,5 __________________________________________ A-5I...
  31. S

    Linear algebra - eigenvalues and eigenvectors and hermitian

    Homework Statement I attached the problem in a picture so its easier to see. Homework Equations The Attempt at a Solution These are the values i got \lambda_ 1 = 1 \lambda_ 2 = -1 x_1 = [-i; 1] (x_1)^H = [i 1] x_2 = [ i; 1] (x_2)^H = [-i 1] * where x_1 and x_2 are...
  32. jinksys

    Computing eigenvalues and eigenvectors

    Find the characteristic polynomial, eigenvectors, and eigenvalues of the matrix. [ 2 -2 3 0 3 -2 0 -1 2 ] My characteristic poly is x^3 - 7x^2 + 14x - 8 The roots are 1, 2, and 4. When solving the system, (2I - A)x = 0 I get: [ 0 1 0 0 0 0 1 0 0 0 0 0 ] Can...
  33. A

    Seeking of eigenvalues and eigenvectors of a given matrix

    Homework Statement in seeking of eigenvalues and eigenvectors of a given matrix A, is it permissible first to simplify A by means of some elementary operation? (that is, are the eigenvalues and eigenvector of A invariant with respect to elementary row operation)? (prove it)Homework Equations...
  34. N

    Why does commuting matrices have same eigenvectors?

    I googled for a proof,but didn't find one. Could anyone give me a link to a proof?
  35. Z

    Eigenvectors from complex eigenvalues

    how does one systematically find the eigenvectors of a 2x2 (or higher) Real matrix given complex eigenvalues?
  36. A

    Eigenvectors & subspace spanning

    The question is at the end of a chapter on spanning vector spaces. Homework Statement Let P denote an invertible n x n matrix. If \lambda is a number, show that E_{\lambda}(PAP^{-1}) = \left\{PX | X\;is\;in\;E_{\lambda}(A)\right\} for each n x n matrix A. [Here E_{\lambda}(A)} is...
  37. S

    Eigenvectors with a repeated eigenvalue

    Homework Statement For the following linear system: \frac{dx}{dt} = -2x \frac{dy}{dt} = -2y Obtain the general solution. Homework Equations The Attempt at a Solution A= -2 0 0 -2 Using the determinant of A-\lambdaI I got a repeated eigenvalue of -2. I am...
  38. T

    Using Eigenvectors to produce a Diagonal matrix

    Homework Statement If A=[{5,3},{-2,-2}], find the eigenvectors of A. Using these eigenvectors as matrix P, find P-1 and thus prove P-1AP is diagonal. Homework Equations None The Attempt at a Solution So i can get the eigenvectors to be <3,-1> and <1,-2> corresponding to eigenvalues 4...
  39. T

    Find the eigenvalues and eigenvectors for the matrix

    Homework Statement Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}] Homework Equations None The Attempt at a Solution Well eigenvalues is easy, and turn out to be 14 and 3. So using eigenvalue 3, the two equations 10x1 + 5x2=0 and 2x1 + x2=0. Using these, I assumed...
  40. C

    Eigenvalues and Normalised Eigenvectors

    Homework Statement I have a matrix H= [h g g h] and I need to find the eigenvalues and normalised eigenvectors Homework Equations The Attempt at a Solution I subtracted lamda from the diagonal and then solved for the determinant equally zero. The eigenvalues I found were...
  41. T

    Linear algebra - Spectral decompositions: Eigenvectors of projections

    Homework Statement Let P1 and P2 be the projections defined on R^3 by: P1(x1, x2, x3) = (1/2(x1+x3), x2, 1/2(x1+x3)) P2(x1, x2, x3) = (1/2(x1-x3), 0, 1/2(-x1+x3)) a) Let T = 5P1 - 2P2 and determine if T is diagonalizable. b) State the eigenvalues and associated eigenvectors of T...
  42. G

    Finding the eigenvectors from complex eigenvalues

    Homework Statement This isn't really a question in particular. I am doing my first Differential Equations course, and in the complex eigenvalues part, I am getting confused as to how to find the eigenvectors. Example: Solve for the general solution of: x' = (1 -1)x (don't know how to...
  43. X

    Eigenvalues and eigenvectors of a 3x3 matrix (principal stresses)[programming]

    I need to compute the 3 eigenvalues and 3 eigenvectors of a symmetric 3x3 matrix, namely a stress tensor, computationaly (in C++). More specific details http://en.wikipedia.org/wiki/Principal_stress#Principal_stresses_and_stress_invariants". Basically 2 questions: 1. I am running into trouble...
  44. A

    Linear algebra, eigenvectors and eigenvalues

    If v is an eigenvector of an invertible matrix A, which of the following is/are necessarily true? (1) v is also an eigenvector of 2A (2) v is also an eigenvector of A^2 (3) v is also an eigenvector of A^-1 A) 1 only B) 2 only C) 3 only D) 1 and 3 only E) 1,2 and 3 I am pretty sure...
  45. M

    Finding the eigenvectors of a 3x3 matrix - help please

    Homework Statement Determine the eigenvalues and eigenvectors of the matric, A: A=\left[\begin{array}{ccc}1 & 1 & 0\\ 1 & -2 & 0\\ 0 & 0 & 1\end{array} Homework Equations I think i understand what is going on. I have found the matrix equation to be...
  46. J

    1 Eigenvalue and 2 Eigenvectors

    Hi all, Let's say we have a symmetric matrix A with its corresponding diagonal matrix D. If A has only 1 eigenvalue, how do we show that there exists 2 eigenvectors? thanks!
  47. C

    Eigenvectors of the hamiltonian

    Homework Statement The Hamiltonian of a system has the matrix representation H=Vo*(1-e , 0 , 0 0 , 1 , e 0 , e , 2) Write down the eigenvalues and eigenvectors of the unperturbed Hamiltonian (e=0) Homework Equations when unperturbed the Hamiltonian will...
  48. A

    How to Express a Vector as a Linear Combination of Eigenvectors?

    Hey guys, I'm studing to my exams now, and I came accors this question i eigenvectors where you find them and bla bla. There is part to it which asks to express vetor X= [2/1] as a linear combination of eigenvectors. Hence calculate B2X, B3X, B4X and B51X, simplifying your answers as...
  49. H

    Bases of common eigenvectors

    Given a set of n<d commuting operators, either degenerate or non-degenerate, in a d-dimensional Hilbert space, is there an effective analytical method of finding an orthonormal basis composed of d eigenvectors common to all the operators in the set? The operators are dxd complex square matrices...
  50. N

    Help finding eigenvectors to simple 2x2 matrix

    Homework Statement Find a fundamental set of real solutions of the system. x'=[-0.5 1 ]x [-1 -0.5] The Attempt at a Solution I calculated the eigenvalues to be r1 = -0.5+i and r2 = -0.5-i Then, attempting to calculate the eigenvectors, I plugged the numbers into the system...
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