What is Eigenvalues: Definition and 849 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. A

    Proof by Induction Question on Determinants and Eigenvalues

    Thanks, although I still haven't managed to factorise the expression although I did type it up in LaTeX! Homework Statement Prove by induction that the following statement is true for all positive integers n. If \lambda is an Eigenvalue of the square matrix A, then \lambda^n is an eigenvalue...
  2. A

    What are the eigenvectors for lambda = -1?

    Homework Statement find the general solution to x'=Ax; where A is a 3x3matrix: A=[0 1 1; 1 0 1; 1 1 0] Homework Equations det(A-lambda*I)=0 The Attempt at a Solution i found the eigenvalues to be 2, -1, -1. for lambda=2 i found the corresponding eigenvector to be a 3x1 martrix...
  3. A

    Subspaces and eigenvalues

    Given a square matrix, if an eigenvalue is zero, is the matrix invertible? I am inclined to say it will not be invertible, since if one were to do singular value decomposition of a matrix, we would have a diagonal matrix as part of the decomposition, and this diagonal matrix would have 0 as an...
  4. M

    Finding Eigenvectors for Distinct Real Eigenvalues

    Homework Statement I don't know how to put matrices in, so I'll just link an http://forum.bodybuilding.com/attachment.php?attachmentid=3339921&d=1305058219" Basically find the solution for that matrix. Homework Equations The Attempt at a Solution This was the...
  5. H

    Eigenvalues and Eigenvectors of Invertible Linear Operators and Matrices

    Let L : V>>>V be an invertible linear operator and let lambda be an eigenvalue of L with associated eigenvector x. a) Show that 1/lambda is an eigenvalue of L^-1 with associated eigenvector x. For this question, the things I know are that L is onto and one to one. Therefore, how to prove this...
  6. D

    Finding eigenvalues of a 3 x 3 matrix

    I have a 3 x 3 matrix A = (0 -1 -3) (2 3 3) (-2 1 1) Let & represent lambda here. I am trying to find the eigenvalues of A. I start off by taking the characteristic equation of A and end up with -&[(&-3)(&-1) -3] + (2& - 8) - 3(-2& + 8) yet can't then get that factored down...
  7. S

    Order of Eigenvectors in Matrix Generation: Does it Make a Difference?

    Homework Statement When generating a matrix from eigenvectors, does it matter in which order the columns are placed?
  8. U

    Eigenvectors eigenvalues and constant of motion

    Homework Statement a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = Hx+Hy+Hz, where Hx=1/2*(px2+x2), Hy=1/2*(py2+y2), Hz=1/2*(pz2+z2) Check that the standard angular momentum operators Lx, is a constant of motion. b.) By knowing that the...
  9. X

    Eigenvalues for a 4th order boundary value problem

    Homework Statement y^{(4)}+\lambda y=0 y(0)=y'(0)=0 y(L)=y'(L)=0 Homework Equations The hint says... let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution Listening to the hint, I got r=\pm\mu With multiplicity 2 of each. So that means.. y=c_1 e^{\mu t}+c_2te^{\mu...
  10. C

    Calculating Eigenvalues of ODE's x1', x2

    x1' = x1 - 5x2 x2' = x1 + 3x2 \begin{bmatrix} 1 & -5\\ 1 & 3\end{bmatrix} \begin{bmatrix} 1-\lambda & -5\\1 & 3-\lambda\end{bmatrix} The eigenvalue I have is lambda = 2+/- 2i. Using lambda = 2-2i, I get the following: \begin{bmatrix} -1+2i & -5\\1 & 1+2i\end{bmatrix} I get an...
  11. T

    Understanding Eigenvectors and Eigenvalues in Linear Algebra

    I am trying to get an eigenvector for the following matrix, I am up to the final step. 4 1 0 0 I got it to be -1 4 is this the same as 1 -4 sorry I am pretty new to linear algebra.
  12. M

    Find Complex Eigenvalues for 3x3 Matrix with All 9 Numbers at .3 | Homework Help

    Homework Statement A 3x3 matrix with all 9 of the numbers being .3 Find all the eigenvalues. Homework Equations The Attempt at a Solution I worked through it and I ended up with (l=lamda) l^3-.9l^2+.54l-.162=0 With my calculator I found one of the values, which means that there...
  13. A

    Group Theory, unitary representation and positive eigenvalues

    Hi, I'm new in this forum. I have a problem i can't solve and searching on Google i couldn't find anything. It says: If D(g) is a representation of a finite group of order n , show that K = \sum^{i=1}_{n} D^{\dagger} (g_i) D(g_i) has the properties: b) All eigenvalues of K are...
  14. F

    Finding Eigenvalues and questions

    Homework Statement http://img703.imageshack.us/img703/4489/unledzh.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Ax = λx Ax = x Ax - x = 0 (A - I)x = 0 I set up my matrix...
  15. F

    What are Eigenvectors and Eigenvalues?

    Homework Statement http://img820.imageshack.us/img820/4874/cah.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Did it already, 3 is the eigenvalue b) This is just finding the nullspace and the basis of the nullspace are my eigenvectors right? c) ignore...
  16. kini.Amith

    Are Matrices with the Same Eigenvalues Always Similar?

    given that 2 matrices have the same eigenvalues is it necessary that they be similar? If not, what is the connection between those 2?
  17. S

    Suppose I get the eigenvalues of A, which are

    Suppose I get the eigenvalues of A, which are \lambda_{1},\lambda_{2},\dots \lambda_{n}. Also, given any polynomial f(x), I get the eigenvalues of f(A). I'm trying to show that the eigenvalues of f(A) are f(\lambda_{1}),f(\lambda_{2}),\dots f(\lambda_{n}). Is this possible? How would I go about...
  18. S

    I am trying to relate eigenvalues with singular values. In particular,

    I am trying to relate eigenvalues with singular values. In particular, I'm trying to show that for any eigenvalue of A, it is within range of the singular values of A. In other words, smallestSingularValue(A) <= |anyEigenValue(A)| <= largestSingularValue(A). I've tried using Schur...
  19. P

    Help with a Problem Involving Eigenvalues and Exponential Functions

    1.\frac{dx}{dt}= \stackrel{9 -12}{2 -1} x(0)=\stackrel{-13}{-5} So I seem to be having issues with this problem There are 2 eigenvalues that I obtained from setting Det[A-rI]=0 That gave me r^{2}-8r+15=0 solving for r and finding the roots i got (r-3)*(r-5)=0 so the...
  20. M

    Eigenvalues of Laplacian with Boundary Condition

    Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.). I said: ∇•(u∇u)=u∆u+∇u2, since ∆u = 0, we have ∇•(u∇u)=∇u2 ∫...
  21. I

    Eigenvalues of Laplacian on parametric surface

    Hello. I would like to numerically determine eigenvalues of a rectangular membrane which is twisted for \frac{\pi}{2}. Example picture: I'm solving Helmholtz equation: \nabla^2u+k^2u=0 where u=u(x,y) and \nabla^2 u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2v}{\partial y^2}...
  22. W

    Eigenfunctions from eigenvalues unsure

    Homework Statement using X''(x)+ lambda*X(x)=0 find the eigenvalues and eigenfunctions accordingly. Use the case lambda=0, lambda=-k2, lambda=k2 where k>0 Homework Equations X(0)=0, X'(1)+X(1)=0 The Attempt at a Solution I know that for lambda=0 X(x)=C1x+C2 which applying the...
  23. B

    Self consistent method for eigenvalues

    Hi all, I am trying to find numerically the eigenvalues of a nonlinear schroedinger equation in a similar way as the Self Consistent Field method for Hatree-Fock problems. Does anybody know in the SCF calculation how to improve the convergency? Is there any trick other than simply inserting...
  24. C

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    Homework Statement y'' + (lambda)y = 0, y'(0) = 0, y(1) = 0 We are told that all eigenvalues are nonnegative. Even with looking at the solution manual, I am unsure how to start setting these up. I've been starting by doing the following: y(x) = A cos cx + B sin dx y'(x) = -Ac sin(cx) + Bd...
  25. jegues

    Eigenvalues Sturm-Liouville system

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  26. S

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  27. F

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    So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand...
  28. R

    Positive definite matrix and its eigenvalues

    I need to know if there is any relationship between the positive definite matrices and its eigenvalues Also i would appreciate it if some one would also include the relationship between the negative definite matrices and their eigenvalues Also can some also menthow the Gaussian...
  29. P

    Eigenvalues of sum of a Hermitian matrix and a diagonal matrix

    Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...
  30. S

    Eigenvalues of Linear Time Varying systems

    The usual eigenvalues of a LTV system does not say much about the stability but my intuition tells me there should be some kind of extension that applies to LTV systems as well. Like including some kind of inner derivative of the eigenvalues or something, I don't know... I guess in some way...
  31. H

    Find Matrix A from eigenvalues and eigenvectors?

    Homework Statement Matrix A has eigenvalues \lambda1= 2 with corresponding eigenvector v1= (1, 3) and \lambda2= 1 with corresponding eigenvector v2= (2, 7), find A. Homework Equations Definition of eigenvector: Avn=\lambdanvn The Attempt at a Solution I tried this by making...
  32. Z

    Eigenvalues of a polynomial transformation

    Homework Statement Let V be the linear space of all real polynomials p(x) of degree < n. If p \in V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? Homework Equations Not sure...
  33. E

    Solving a PDE Eigenvalue Problem: Proving All Eigenvalues Are Positive

    I have a PDE test next week and I'm kinda confused. How do you prove that eigenvalues are all positive? I know Rayleigh Quotient shows the eigenvalues are greater than or equal to zero, but can someone explain the next step. Thanks in advance
  34. U

    Eigenvalues for a Hamiltonian

    Homework Statement Consider the Hamiltonian \hat{}H = \hat{}p2/2m + (1/2)mω2\hat{}x2 + F\hat{}x where F is a constant. Find the possible eigenvalues for H. Can you give a physical interpretation for this Hamiltonian? Homework Equations The Attempt at a Solution I don't think...
  35. C

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  36. D

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    Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen.jpg The Attempt at a Solution It is a bit too long to type it all out, but I was wondering whether I am correct: I got, A = 7/2 , B = 0 , C = -1/8 , D = 1/8 And from this I worked out, x2(1) =...
  37. J

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  38. N

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

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  39. 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??
  40. L

    Quantum Mechanics. Normalised basis wave functions and Eigenvalues.

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  41. L

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  42. X

    Eigenvalues and Norms: Showing Existence of a Nonsingular Matrix

    Homework Statement Let A \in \mathbb{C}^{n \times n} and set \rho = \max_{1 \le i \le n}|\lambda_i|, where \lambda_i \, (i = 1, 2, \dots, n) are the eigenvalues of A. Show that for any \varepsilon > 0 there exists a nonsingular X \in \mathbb{C}^{n \times n} such that \|X^{-1}AX\|_2 \le...
  43. Telemachus

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  44. 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)...
  45. 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?
  46. Saladsamurai

    Eigenvalues, Eigenspaces, and Basis

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  47. M

    Matrix Similarity and Eigenvalues

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  48. M

    Eigenvalues for integral operator

    Homework Statement Find all non-zero eignvalues and eigenvectors for the following integral operator Kx := \int^{\ell}_0 (t-s)x(s) ds in C[0,\ell] Homework Equations \lambda x= Kx The Attempt at a Solution \int^{\ell}_0 (t-s)x(s) ds = \lambda * x(t)...
  49. L

    What is the issue with calculating eigenvalues using rgg.f?

    Hey folks, I'm having an issue using a routine from the netlib that is supposed to calculate eigenvalues and eigenvectors. The canned routine can be found here: http://www.netlib.org/seispack/rgg.f I want to find the eigenvalues of a matrix (a more complex hamiltonian), so for my simple...
  50. A

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

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