What is Eigenvalues: Definition and 851 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. 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...
  2. 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...
  3. C

    Shape Operators and Eigenvalues

    This is probably falls within a problem of Mathematica as opposed to a question on here but I have a question about the following: Given some cylinder with the shape operator matrix: {{0,0},{0,-1/r}} We get eigenvalues 0 and -1/r and thus eigenvectors {0, -1/r} and {1/r, 0} by my...
  4. H

    System of differential equations eigenvalues

    Homework Statement solve the system: dx/dt = [1 -4] x _______[4 -7] with x(0) = [3] __________[2] Homework Equations The Attempt at a Solution I got both eigenvalues of the matrix are -3 and so both eigenvectors are [1]...
  5. 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...
  6. T

    Quantum Mechanics Operators, Hermitian and Eigenvalues

    1. a) The action of the parity operator, \Pi(hat), is defined as follows: \Pi(hat) f(x) = f(-x) i) Show that the set of all even functions, {en(x)}, are degenerate eigenfunctions of the parity operator. What is their degenerate eigenvalue? The same is true for the set of all odd functions...
  7. L

    Matrices with all zero eigenvalues

    If I have a matrix for which all eigenvalues are zero, what can be said about its properties? If I multiply two such matrices, will the product also have all zero eigenvalues? Thanks
  8. L

    Does the Matrix xyTA Have More Than One Non-Zero Eigenvalue?

    We have vectors x,y of size n and a matrix A of size nxn. Is it true that the matrix xyTA has at most one non zero eigenvalue? Why is it so?
  9. G

    Commuting matrices have common eigenvalues

    Homework Statement How do we prove that commuting matrices have common eigenvalues? Homework Equations The Attempt at a Solution
  10. T

    Finding eigenvalues, Shankar exercise 1.8.3

    First, I appologise if this is in the wrong place, while the book is QM, the question is pure maths. Also I'm not sure if this techically counts as homework as I am self studying. Finally, sorry for the poor formatting, I'm not that good with LaTeX Homework Statement Given the matrix...
  11. N

    Can a 3x3 matrix have 4 eigenvalues?

    Homework Statement Prove or disprove the title of this thread. Homework Equations AX=(lamda)X The Attempt at a Solution I don't know where to start
  12. Shackleford

    Show eigenvalues of hermitian operator are real

    http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165642.jpg?t=1287612122 http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165727.jpg?t=1287612136 Thanks.
  13. A

    All Eigenvalues Lie on the Unit Circle

    Hi everyone Consider a 2x2 partitioned matrix as follow: A = [ B1 B2 ; B3 B4 ] I'm sure that all eigenvalues of A are on the unit circle (i.e., abs (all eig) = 1 ). but, I don't know how to prove it. Is there any theorem?
  14. B

    Are the Eigenvalues of the Zero Ket Always Zero?

    Homework Statement I am wondering if I can make the sweeping generalization that the eigenvalues of the zero ket are zero. I further generalize that the zero ket is not of interest, as far as physical observables occur. Homework Equations the eight axioms of vector spaces...
  15. D

    Finding eigenvalues and eigenspaces with only this info

    Lets say I have a 3x3 matrix 'A' and one known eigenvalue 'z' and one known eigenvector 'x', but they don't "belong" to each other, as in Ax =/= zx Is there a way of finding the other eigenvalues and eigenspaces of A using only this piece of information? Thanks.
  16. B

    Finding Eigenvalues for u''+λu=0

    Hi guys, Can someone please explain how you find the eigenvalues of this type? u''+\lambda u =0 or point me to some decent literature? regards Brendan
  17. J

    Simple quantum problem - find eigenvalues, probabilities, expectation value?

    hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...
  18. G

    Continuous Eigenvalues: QM Position & Momentum Operators Explained

    Dear all, in basic QM books the position and momentum operators (continuous eigenvectors) are introduce by means of the dirac delta and some analogies are made with the infinite dimensional, but discrete case in order to provide some intuition for the integral formulas presented. My knowledge...
  19. 4

    Using software to solve eigenvalues of Hamiltonian

    This is my first time posting in this forum, or any, so I'm sorry if something is out of place. I'm doing undergrad research with a professor on quantum supercomputing and I need to use some software to find the eigenvalues of the energy using the Hamiltonian. He suggested I used maplesoft...
  20. J

    Energy eigenvalues and ground-state energy

    Homework Statement The energy eigenvalues of a particles of mass, m, confined to a 3-d cube of side a are: E_{nx,ny,nz}=\frac{a(n^{2}_{x}+n^{2}_{y}+n^{2}_{z})}{b}+ Vo where: a= planks constant^2(pi)^2 b=2m^2 nx,ny,nz = any positive integers. What are the ground-state kinetic and potential...
  21. P

    Understanding Eigenvectors and Eigenvalues in Linear Transformations

    Can someone explain these to me?
  22. A

    Eigenvalues of AX - XA: Finding Eigenvectors for a Real 2x2 Symmetric Matrix

    Hi all, Here is this problem that I have been at for some time now: find eigenvalues and corresponding eigenvectors of the following linear mapping on a vector space of real 2 by 2 matrices: L(X) = AX - XA, where A is 2 by 2 symmetric matrix that is not a scalar multiple of identity...
  23. S

    On number of negative eigenvalues of a matrix

    Homework Statement When trying to solve a question about parameter independence of certain aspects of the Jacobian of a real valued function on a manifold I came to the point where I have to show the following: Let A be a matrix, J be the Jacobian of an orthogonal transformation (I suppose we...
  24. 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...
  25. 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...
  26. 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...
  27. 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...
  28. Z

    Eigenvectors from complex eigenvalues

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

    Find the eigenvalues of this matrix

    Homework Statement consider the system x' = \left[-1 & -1\\ -.5 & -1\right]x (I'm sorry I can't seem to get a new row in! the second line is [-.5 -1] solve the system. What are the eigenvalues of the coefficient matrix? Classify the equilibrium point at the origin as to type...
  30. S

    Accurate estimation of complex eigenvalues

    Hello, I use Arnoldi iterative algorithm in order to compute the eigenvalues of a matrix. I know that the eigenvalues are of the form \lambda(1+j/c) and I can totally estimate them. The problem that occurs is that both the range of \lambda_0 and c is for example [100,1000]. That means that there...
  31. S

    Mathematica: Eigenvalues for a large symbolic matrix

    I'm trying to compute the eigenvalues for a 32x32 symbolic matrix (with one variable) in Mathematica. I get the following error: Eigenvalues::eival: Unable to find all roots of the characteristic \ polynomial. >> What could be a possible way to proceed? Thanks, Schez
  32. pellman

    Discrete vs continuous eigenvalues

    What determines whether an operator has discrete or continuous eigenvalues? Energy and momentum sometimes have discrete eigenvalues, sometimes continuous. Position is always continuous (isnt it?) Spin is always discrete (isn't it?) Why?
  33. M

    What is the meaning of T(1), T(x), and T(x2) in polynomial transformations?

    Homework Statement T: R3[x] R3[x] // for some reason the arrow symbol isn't working! When I do the arrow it previews as the third power for some reason. Also, whenever I preview post, it adds [b]1 [b]2 b[3] again for some reason and I have to delete those lines every time...a bit fustrating...
  34. P

    Non-zero eigenvalues and square matrix

    hello, two quick question here. I've got the answer correct (I think), but I am not too sure how to explain it in words. So I hope someone tell me is my attempted explanation correct. 1) what is the maximum of non-zero eigenvalues a singular square matrix with 7 rows can have? up to...
  35. 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...
  36. G

    Eigenvalues of a 5x5 Matrix, continued

    Homework Statement The https://www.physicsforums.com/showthread.php?t=403476" was to determine the eigenvalues of the following matrix. The problem of interest deals with actually finding a solution to the system above without the use of matrix methods. Homework Equations The...
  37. G

    How can you solve for the remaining variables when multiplied by zero?

    Homework Statement The Attempt at a Solution I haven't tackled anything bigger than a 3x3 matrix. Anyone have any good pointers for reducing this matrix? I'm assuming the quickest way is still going to be the cofactor method?
  38. T

    Help Finding eigenvalues of angular momentum operators

    urgent help!.. Finding eigenvalues of angular momentum operators the question is asking to find the eigenvalues of: 3/5 Lx - 4/5 Ly ... I feel that i have to connect it with the L^2 and Lz operators but i just have no idea how to start .. any suggestions will be greatly appreciated ..
  39. J

    Distinct Eigenvalues: Can Zero be an Eigenvalue?

    Can zero be a distinct eigernvalue?
  40. G

    Systems of ODE's double-zero eigenvalues

    Homework Statement I put a triangle around the problem of interest. Homework Equations The Attempt at a Solution I solved for the eigenvalues, resulting in double-zero values. My question is, using the variation of parameters method, which is what (14) refers to in the...
  41. B

    Eigenvalues of a reduced density matrix

    My lecturer keeps telling me that if a density matrix describes a pure state then it must contain only one non-zero eigenvalue which is equal to one. However I can't see how this is true, particularly as I have seen a matrix \rho_A = \begin{pmatrix} 1/2 & - 1/2 \\ -1/2 & 1/2 \\ \end{pmatrix} for...
  42. 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...
  43. A

    Eigenvector with Complex Eigenvalues - What am I doing wrong?

    Homework Statement Homework Equations Conjugate of a complex number Matrix reductionThe Attempt at a Solution My attempt is bordered. Sorry about the quality. So I'm not sure what I'm missing. I use the exact same method that I use for normal eigenvectors, just with complex numbers in the mix.
  44. G

    Systems of ODE's - Complex Eigenvalues

    Homework Statement Find the general solution of the given system. The given matrix is X' = (1st row (1,-1,2) 2nd row (-1,1,0) 3rd row (-1,0,1))X 2. The attempt at a solution The eigenvalue determinant = (1st row (1-λ,-1,2) second row (-1,1-λ,0) 3rd row (-1,0,1-λ) Solving the...
  45. J

    Simplifying a solution that has complex eigenvalues

    Homework Statement I'll give an example. Ex: x'=[-1/2 1; -1 -1/2]x. Homework Equations Assume a solution of the form x=$ert for these type of problems. Euler's formula: ebi = cosb + isinb The Attempt at a Solution |A-rI|=0 ---> r= -1/2 +/- i ---> x= e-t/2 ( C1(cost...
  46. D

    Trace, determinant, and eigenvalues 3x3

    Use the trace and determinant to compute eigenvalues. I know how to do this with a 2x2 but not sure how to do it with a matrix of nxn where n>2. \begin{bmatrix} \frac{1}{2} & \frac{1}{3} & \frac{1}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}...
  47. S

    Proving eigenvalues = 1 or -1 when A = A transpose = A inverse A is circulant

    Homework Statement Prove all eigenvalues = 1 or -1 when A is circulant and satisfying A=A^T=A^-1 I can think of an example, the identity matrix, but i can't think of a general case or how to set up a general case. Homework Equations The Attempt at a Solution I can only show by...
  48. J

    Repeated Eigenvalues: How to Solve for a General Solution

    This problem, and all the others on this homework assignment, are making me angry. Homework Statement Find the general solution of the system of equations. ... x'=[-3 5/2; -5/2 2]x Homework Equations Just watch me solve The Attempt at a Solution Assume there's a...
  49. K

    Eigenfunctions and Eigenvalues

    Hi, I am having a lot of difficulty conceptually understanding what eigenfunctions and eigenvalues actually are, their physical meaning, i.e. what they represent, and how they interact. Would anybody happen to be able to explain them in relatively simple terms? I didn't know whether to put...
  50. J

    Solving a system of diffy q's with complex eigenvalues

    Homework Statement Express the general solutoins of the system of equations in terms of real-valued functions. x'= [1 0 0; 2 1 -2; 3 2 1]x (I wrote the matrix MATLAB-style) Homework Equations The coolest equation ever: eib=cosb + isinb The Attempt at a Solution...
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