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. S

    Matrix with repeated eigenvalues

    The question is asking for what values of x will the matrix have at least one repeated eigenvalue (algebraic multiplicity of 2 or greater). The matrix is | 3 0 0 | | 0 x 2 | So naturally a normal attempt to find the eigenvalue in a question with only intergers | 0 2 x | I would continue...
  2. M

    Proving Hermitian if it has real eigenvalues

    If you had an operator A-hat whose eigenvectors form a complete basis for the Hilbert space has only real eigenvalue how would you prove that is was Hermitian?
  3. M

    Find the eigenvalues and a basis of each eigenspace

    Homework Statement Matrix A is -4 4 4 -4 4 4 4 -4 -4 It has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigenspace. The Attempt at a Solution I got the eigenvalues: the one of multiplicity 1 is -4 the one of multiplicity...
  4. J

    Orthogonality of eigenfunctions with continuous eigenvalues

    Homework Statement With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1. 2. The attempt at a solution The first...
  5. S

    Comparing Eigenvalues for Sarah & Janie: Is It Compatible?

    Suppose for a given matrix A, Sarah finds the eigenvectors v1 = [1 3 4 5]' and v2 = [5 6 3 4]' form a base for eigenspace of labmda = 2. Now suppose Janie finds the eigenvectors v3 = [1 2 2 3]' and v4 = [7 8 7 6]' form a base for eigenspace of lambda = 4. Is Janie's solution compatible with...
  6. S

    Distinct Eigenvalues in matlab

    All I want to know is if a number and its negative appear as eigenvalues of a matrix, are they considered distinct? I have 4,1,-2,3 and -1 as eigenvalues of a particular matrix, but I can't get 5 linearly independent eigenvectors (to diagonalise the original matrix). I've plugged away and...
  7. K

    What Are the Eigenvalues and Degeneracies of a Coupled Ising Spin System?

    Homework Statement Consider two Ising spins coupled together −βH = h(σ1 + σ2) + Kσ1σ2, where σ1 and σ2 commute and each independently takes on the values ±1. What are the eigenvalues of this Hamiltonian? What are the degeneracies of the states? The Attempt at a Solution Four possible...
  8. W

    Orthogonal matrix and eigenvalues

    a) Let M be a 3 by 3 orthogonal matrix and let det(M)=1. Show that M has 1 as an eigenvalue. Hint: prove that det(M-I)=0. I think I'm supposed to begin from the fact that det(M)=1=det(I)=det(MTM) and from there reach det(M-I)=0 which of course would mean that there's an eigenvalue of 1 as...
  9. maverick280857

    Diagonalizing a square matrix with degenerate eigenvalues

    Hi This is more of a math question but in the context of Quantum Mechanics, hence I posted it here. Suppose I have a matrix A of order 3x3 with three eigenvalues: 0, 0, 5. I am supposed to find the diagonalizing matrix for A. I know that in general, if P denotes the matrix of eigenvectors of...
  10. F

    Iterated maps and eigenvalues and vectors

    Totaly stuck on this one can't even start to fathom an attempt. first part of the question is show that the eigenvalues of the matrix (2x2 left to right) 4,-1,-4,4 are sigma1=2 and sigma2 =2 and eigenvectors e1= (1,2)T and e2=(1,-2)T done this no problem but am writing this as the second part of...
  11. S

    Eigenvalues & eigenvectors of N x N matrix?

    How to get eigenvalues & eigenvectors of N x N matrix? Please can anyone help me out i have searched a lot but not able to find the solution. Regards
  12. J

    Eigenvalues of A: Same Eigenspaces for A^-1, Transpose, A^k

    Is this a correct realization? The eigenspaces corresponding to the eigenvalues of A are the same as the eigenspaces corresponding to the eigenvalues of A^-1, transpose of A, and A^k for any k > 1. It took me some time to realize this but the v's, when you manipulate these equations, don't...
  13. rocomath

    Understanding Eigenvalues: Question on Diagonalizability and Orthogonality

    I wasn't sure about this question on my exam. Let eigenvalues be 0, 1, -1 Is it true that a. Not Invertible b. Diagonalizable c. Orthogonal a. True, since the determinant is 0 b. I'm not sure, but I chose False b/c I had an eigenvalue of 0 c. True, ah not sure either LOL!
  14. rocomath

    What is the significance of Eigenvalues in a 2 by 2 matrix?

    Assuming 2 by 2. Ok, I'm asked to find the Eigenvalues. How do I know which should be lambda 1 and lambda 2? I can find the lambda's easily, but does it matter which is 1 or 2? It becomes important when I'm asked to diagonalize. A=S\Lambda S^{-1}
  15. A

    I don't get Eigenvalues or Eigenvectors

    I just finished Differential Equations, and I know how to find eigenvalues/eigenvectors, and I understand how to use them to solve a differential equation. But I don't really understand "what they are". How is a matrix with complex eigenvalues any different than a matrix with real...
  16. A

    Approximate Eigenvalues for [4 3] [3 5]

    [SOLVED] Approximate eigenvalues Homework Statement Use some QR method to approximate the eigenvalues of [4 3] [3 5] and compare with the actual values. The actual values are (9±√37)/2 Homework Equations A(0)=Q(0)R(0) A(1)=R(0)Q(0) A-α(0)I=Q(0)R(0) A(1)=R(0)Q(0) + α(0)I...
  17. A

    Positive Definite Matrices eigenvalues

    [SOLVED] Positive Definite Matrices a) If A is Symmetric show that A-λI is positive definite if and only if all eigenvalues of A are >λ, and A-λI is negative definite if and only if all eigenvalues of A are <λ. b) Use this result to show that all the eigenvalues of [ 5 2 -1 0] [ 2 5 0...
  18. A

    Can QR Method Accurately Approximate Eigenvalues? [SOLVED]

    [SOLVED] Approximate eigenvalues Use some QR method to approximate the eigenvalues of [4 3] [3 5] and compare with the actual values. The actual values are (9±√37)/2
  19. I

    Interpretation/significance of the eigenvalues for a system?

    Hi folks! I wasn't sure where to put this... so I put it here! I'm wondering if there is a physical interpretation/significance of the eigenvalues for a system? I've had people tell me things like "they're the basic solutions to the system" but I just don't quite see what they're saying...
  20. F

    Eigenvalues and diagonalizability

    This is a concept question.. I'm having trouble understanding why for an n x n matrix A, in order to have eigenvalues, it must have linearly dependent columns (so that a nontrivial solution exists), but for the same A, in order to be diagonalizable, the columns must be linearly INdependent...
  21. H

    Solving Repeated Eigenvalues - [2,2,1;1,3,1;1,2,2]

    Homework Statement Obtain the eigenvalues and corrosponding eigenvectors for the matrix: [2,2,1;1,3,1;1,2,2] Homework Equations The Attempt at a Solution I can solve for the eigenvalues, 5, 1, and 1 I can solve the eigenvalue 5 for the eigenvector B[1;1;1] Yet somehow, the...
  22. S

    What does multiplicity in eigenvalues mean?

    Can someone please explain multiplicity to me? I've been able to solve the problems involving it, but I'm not quite sure what it means in terms of the eigenvalue. Thanks.
  23. O

    MATLAB, eigenvalues and condition number of a symmetric square matrix

    2. Write a MATLAB® function to calculate the condition number of a symmetric square matrix of any size by means of Eigenvalues: § The power method should be used to calculate the Eigenvalues. § The script (function) should give an error message if the matrix is not...
  24. E

    Eigenvalues of the operator x d/dx

    a) Consider the operator x d/dx(where 1st d/dx acts on the function, then x acts on the resulting function by simply multiplying by x )acting on the set of functions of a real variable x for x>0. What are the eigenvalues and the corresponding eigenfunctions of this operator? b) What about...
  25. E

    Eigenfunctions & Eigenvalues

    a) Consider a linear operator L with 2 different eigenvalues a1 and a2, with their corresponding eigenfunction f1 and f2. Is f1 + f2 also an eigenfunction of L? If so, what eigenvalue of L does it correspond to? If not, why not? b) Answer the same question as in part (a) but for the...
  26. K

    Local Extrema, Hessian & Eigenvalues

    1) f(x,y,z)=x3-3x-y3+9y+z2 Find and classify all critical points. I am confused about the following: The Hessian matrix is diagonal with diagonal entries 6x, -6y, 2. Now, the diagonal entries of a diagonal matrix are the eigenvalues of the matrix. (this has to be true, it is already...
  27. H

    Understanding Eigenvalues in Matrix Construction from Differential Equations

    Let's say that I have to construct a 2 X 2 matrix from a second-order differential equation, turning it into a system of first order linear equations, and find its eigenvalues. I'll have two variables that correspond to the two columns in the matrix. If I swap columns, I end up with two...
  28. R

    How to Prove the Second Part of a Multi-Part Question on Complex Eigenvalues?

    Homework Statement This is the second part of a multi-part question. Part (a) shows that: x'' = Ax = \left(\stackrel{-2}{4/3}\stackrel{3/2}{-3}\right)x Part (b): Assume x = \epsilone^{rt} and show that (A - r^{2}I)\epsilon = 0 x is the solution to the second order differential equation...
  29. S

    How can diagonal matrices help solve eigenvalue problems?

    hi, can some one give me any hints how to solve this problem? thank you i tried to type it here but it dint come up so i uploaded http://tinypic.com/view.php?pic=2hgtqoz&s=3" with the problem. Thank you so much Recall that for an nxn matrix A with distinct eigenvalues \lambda...
  30. S

    Linear Algebra - Eigenvalues / Eigenvectors

    I have two problems here; one I think I almost have but I'm stuck, and the other I'm pretty much stumped on.Homework Statement Suppose V is a complex vector space and T is in L(V). Prove that T has an invariant subspace of dimension j for each j = 1, ... dim(V). Homework Equations The Attempt...
  31. M

    Given S, T, prove that ST and TS have the same eigenvalues.

    I need help starting/doing this proof. Suppose S,T are Linear Operators on a Finite Dimensional Vector Space V. Prove that ST and TS have the same eigenvalues. A linear operator is a linear map from a vector space to itself. thanks.
  32. S

    Lin. Algebra - Find Eigenvectors / eigenvalues

    Homework Statement Define T in L(F3) by T(z1, z2, z3) = (2*z2, 0, 5*z3). Find all eigenvalues and eigenvectors of T.Homework Equations The Attempt at a Solution Well, since we want to find all the eigenvalues, we want the following equation to hold: T(z1, z2, z3) = (2*z2, 0, 5*z3) = \lambda(z1...
  33. B

    Calculating the determinant of a quadratic for Eigenvalues

    Homework Statement Find the eigenvalues and eigenvectors of A = \left(\begin{array}{ccc}5&1&1\\1&3&1\\1&1&3\end{array}\right) The Attempt at a Solution The problem I'm having is finding the eigenvalues for the matrix. In 2d matricies it's not too bad, but in 3d the...
  34. A

    Diagonalization, eigenvectors, eigenvalues

    [SOLVED] diagonalization, eigenvectors, eigenvalues Homework Statement Find a nonsingular matrix P such that (P^-1)*A*P is diagonal | 1 2 3 | | 0 1 0 | | 2 1 2 | Homework Equations I am at a loss on how to do this. I've tried finding the eigen values but its getting me...
  35. J

    Show that the eigenvalues of the overlap matrix are positive

    Homework Statement The task is to show that the eigenvalues of overlap matrix \tilde S are positive. Homework Equations The overlap matrix is defined as (\tilde S)_{nm} = \langle \xi_n \vert \xi_m \rangle , with \xi_k being the base vectors of the wavefunction...
  36. D

    Partial DIfferential Equations / Eigenvalues

    Hi, I'm barely a high school senior who is somewhat overwhelmed by a univ. course. Anyway, we are just learning to solve some basic PDEs using the method of separation of variables. With this method (and the questions we are given) we check three cases to find the eigenvalues of Sturm-Liouville...
  37. N

    How Can You Create a Math Problem Involving Eigenvalues and Matrix Powers?

    Hi, I'd like to make my own mathematical problem in which you'd have to use the calculation of eigenvalues, eigenvectors and a (high) power of a matrix A. (with definitions: AX = \lambdaX & A^{n} = BD^{n}B^{-1}). I'm not searching for something too complicated, it's more like to integrate...
  38. N

    Eigenvalues and -vectors in class

    Hi, For math we were assigned a subject which we'd present during one class' hour in a group. My group got "Eigenvalues & eigenvectors". So basically first I have to give the definition and explain what it actually is (AX = \lambdaX) and then we can spend the rest of the 45 min on making class...
  39. B

    Eigenvalues and Orthogonal Matrices: Proving Properties Without Prefix

    Question 1 Let A be an nxn matrix such that (A-I)^{2}=O where O is the zero matrix Prove that if {\lambda} is an eigen value of A then {\lambda}=1 My attempt If (A-I)^{2}=O then A=I (1) if {\lambda} is an eigen value of A then Ax={\lambda}x (2) replace (1) in (2) Ix={\lambda}x , but...
  40. B

    EigenValues & EigenVectors proofs

    Question 1: Proove that if λ is an eigenvalue of [A], then 1/λ is an eigenvalue of [A]{T} Question 2 Proove that a square matrices [A] and [A]T have the same Eigenvalues. Question 3: Show that |det(A)| is the product of the absolute values of the eigenvalues of [A]...
  41. P

    Eigenvalues of a tridiagonal matrix

    Could some one please explain the logic behind the subroutine tqli(d,e,z) found on pg.1228 of numerical recipes in fortran. For this subroutine, d and e are one dimensional arrays and z is an optional multi-dimensional array(only used if also seeking for eigenvectors of matrix). The 3 modules...
  42. I

    Finding Eigenvectors & Eigenvalues of A Matrix

    Homework Statement Use direct multiplication to show that for each of the following matrices A, the given vectors v1, v2, and v3 are eigenvectors of A and to find the eigen values lama1, lama2, and lama3 of A: A=top row: (2 -1 3) second row: (-1 6 -1) third row: (3 -5 2)...
  43. P

    Bessel Functions / Eigenvalues / Heat Equation

    Hello Trying to calculate and simulate with Matlab the Steady State Temperature in the circular cylinder I came to the book of Dennis G. Zill Differential Equations with Boundary-Value Problems 4th edition pages 521 and 522 The temperature in the cylinder is given in cylindrical...
  44. C

    Eigenvalues and vectors - finding original matrix

    How do I determine what the original matrix was that yielded these two eigenvalues with the corresponding eigenvectors: \lambda_1 = -3 Eigenvector: [0,1] \lambda_2 = 2 Eigenvector: [1,0] I've played around with det(A-lambda I) but can't find the matrix! I even just did some trial and...
  45. K

    Eigenvalues of a unitary matrix

    Q: Prove htat if a matrix U is unitary, then all eigenvalues of U have absolute value 1. My try: Suppose U*=U^-1 (or U*U=I) Let UX=(lambda)X, X nonzero => U*UX=(lambda) U*X => X=(lambda) U*X => ||X||=|lambda| ||U*X|| => |lambda| = ||X|| / ||(U^-1)X|| And now I am really stuck and...
  46. C

    Complex Numbers: Eigenvalues and Roots

    [SOLVED] Complex Numbers: Eigenvalues and Roots Below are some problems I am having trouble with, the computer is telling me my answers are wrong. It may be the way I am inputting the numbers but as my final is in a week and a half I would like to be sure. Thanks,
  47. G

    Finding Eigenvalues of Real Non-Symmetric Matrices

    Hi, Can somebody provide a link(other than numerical recipes) where I can get an optimized 'C' program for calculating eigenvalues of real nonsymmetric martix. Thanks
  48. R

    Eigenvalues and Eigenspinors: How to find them for the S_y matrix?

    [SOLVED] Eigenvalues and Eigenspinors Homework Statement (a) Find the eigenvalues and eigenspinors of S_{y}. Homework Equations \hat{Q}f(x) = \lambda f(x) The Attempt at a Solution The above equation wasn't given specifically for this problem; but that's the one I'm trying to...
  49. C

    Are there legal ways to quickly find eigenvalues of an operator?

    If I have an operator of the form 1+3\vec{e}\cdot\vec{\sigma} where \vec{e}\cdot\vec{e}=1. How can I find the eigenvalues quickly?
  50. pellman

    Can a Quantum State Have Time-Dependent Eigenvalues?

    Given an operator \hat{Q} (in the Schrodinger picture) in non-relativistic quantum mechanics and a state |\psi(t)\rangle such that \hat{Q} |\psi(t)\rangle=q(t)|\psi(t)\rangle where q(t) is explicitly time-dependent, can we properly say that |\psi(t)\rangle is an eigenstate of Q with a...
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