Eigenvalues Definition and 853 Threads

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. 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...
  2. 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...
  3. A

    Itzykson-Zuber Integral & Degenerate Eigenvalues

    It seems that according to the formula, when the eigenvalues of one of the matrices inside the trace of the IZ integral are degenerate, the integral diverges. Is it correct or the formula is different for this case? For instance, suppose the group is U(N) and I want to calculate \int dU \...
  4. 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...
  5. 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...
  6. 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...
  7. G

    Is C Diagonalizable Given Its Eigenvalue Multiplicity?

    Let C be a 2 × 2 matrix such that x is an eigenvalue of C with multiplicity two and dimNul(C − xI) = 1. Prove that C = P |x 1|P^−1 |0 x| for some invertible 2 × 2 matrix P. I'm not sure where to start EDIT |x 1| |0 x| is the matrix I don't know why it's...
  8. 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...
  9. H

    Bessel Functions - Eigenvalues + Eigenfunctions

    Homework Statement I'm given a standard form of Bessel's equation, namely x^2y\prime\prime + xy\prime + (\lambda x^2-\nu^2)y = 0 with \nu = \frac{1}{3} and \lambda some unknown constant, and asked to find its eigenvalues and eigenfunctions. The initial conditions are y(0)=0 and...
  10. N

    Eigenvalues and density of states

    Hi guys I have an analytical expression f(x) for my density of states, and I have plottet this. Now, I also have a complete list of my Hamiltonians eigenvalues. When I make a histogram of these eigenvalues, I thought that I should get an exact (non-continuous) copy of my plot of f(x). They...
  11. B

    Linear operators, eigenvalues, diagonal matrices

    So I have a couple of questions in regards to linear operators and their eigenvalues and how it relates to their matrices with respect to some basis. For example, I want to show that given a linear operator T such that T(x_1,x_2,x_3) = (3x_3, 2x_2, x_1) then T can be represented by a diagonal...
  12. N

    Degenerate Eigenvalues and Eigenvectors: Understanding Differences in Solutions

    Homework Statement Please see the attached image. The first line just finds the eigenvalues of that matrix. The second line finds the eigenvectors. The third line just takes row 1 and row 3 of that matrix and find the determinant. The fourth line just takes row 2 and row 4 of that...
  13. A

    SOLUTION Multivariate SDE with repeated Eigenvalues

    I am looking forward the solution of multivariate Ornstein–Uhlenbeck differential stochastic equation with repeated eigenvalues. In particular with dy=A(y-c)dt +DdW y is a vector nx1 A is nxn matrix with repeated eigenvalues c is vector of nx1 of constant D is a nxm matrix of...
  14. N

    Can the Eigenvalues of A Determine if A^3=A?

    Hey, I'm wondering if I have a known set of eigenvalues (-1, +1, 0) for A, if I can prove that the matrix A = A3? I can prove that if A3 = A, that the eigenvalues would be −1, +1, and 0. The following is the proof: A*k=lambda*k A3*k=lambda3*k Since A=A3, A*k=A3*k lambda*k=lambda3*k...
  15. P

    Find the eigenvalues of this endomorphism of R[X]

    Homework Statement f is an endomorphism of Rn[X] f(P)(X)=((aX+b)P)' eigenvalues of f? Homework Equations (a,b)<>(0,0) The Attempt at a Solution If a=0, then f(P)=bP', and only P=constant is solution if a<>0, then I put Q=(ax+b)P, f(P)=cP is equivalent to (ax+b)Q'=Q (E)...
  16. K

    Eigenvalues, linear transformations

    Homework Statement T: V-> V, dimV = n, satisfies the condition that T2 = T 1. Show that if v \in V \ {0} then v \in kerT or Tv is an eigenvector for eigenvalue 1. 2. Show that T is diagonalisable. Homework Equations The Attempt at a Solution I have shown in an earlier part...
  17. K

    Eigenvalues and diagonalisation of differentiation as a linear transformation

    Homework Statement Let V be the space of polynomials with degree \leq n (dimV=n+1) i. Let D:V->V be differentiation, i.e. D: f(x) -> f'(x) What are the eigenvalues of D? Is D diagonalisable? ii. Let T be the endomorphism T:f(x) -> (1-x)2 f''(x). What are the eigenvalues of T? Is...
  18. D

    Finding A to the power of n without using eigenvalues

    Oh it gives me headache... been thinking on this problem for a while, and don't even know where to begin! Could anyone give me a hint at least?? :( Problem: Let A be (3x3) matrix : [ 4 -2 2; 2 4 -4; 1 1 0] and u (vector) = [1 3 2]. a) Verify that Au = 2u I got this one without a problem...
  19. G

    What are the eigenvalues and eigenfunctions for T(f(x)) = 5f(x) on C for V:R->R?

    Homework Statement T(f(x)) = 5 f(x) T is defined on C. Find all real eigenvalues and real eigenfunction. V:R -> R Homework Equations Not sure. The Attempt at a Solution No, clue. I can find eigenvalues for matrices, that's not a problem. I'm having problem that its a T(function) =...
  20. M

    Eigenvalues and eigenvectors of the momentum current density dyadic

    Homework Statement What are the eigenvalues and eigenvectors of the momentum current density dyadic \overleftrightarrow{T} (Maxwell tensor)? Then make use of these eigenvalues in finding the determinant of \overleftrightarrow{T} and the trace of \overleftrightarrow{T}^2 Homework...
  21. C

    What happens to the eigenvalues when a constant is multiplied to a matrix?

    Hello, Let's say I have a 2x2 matrix,we call it A with the eigenvalues +1 , -1. Now I let's define that m=m0*A. (m0 is const). Are the eigenvalues become +m0 and -m0? If so why?
  22. matt_crouch

    Is Lambda = 1 an Eigenvalue of the Given Matrix?

    Homework Statement Show that lambda = 1 is an eigenvalue of the matrix 2,-1, 6 3,-3, 27 1,-1, 7 and find the eigenvalues and the corresponding eigenvectors Homework Equations The Attempt at a Solution I don't understand how to actually get eigenvalues and...
  23. S

    Matrices and eigenvalues. A comment in my answer.

    Homework Statement Hello and thanks again to anyone who has replied my posts. Your help is a great deal and really appreciated. I have the following homework question which I have answered and I want a comment if it is valid or illogical: We are given a matrix, with eigenvalues 3 and...
  24. F

    Eigenvalues of hermitian matrix

    1. Let AH be the hermitian matrix of matrix A, and how the eigenvalues of AH be related to eigenvalues of A? [b]3. what I have done is equation no.1: (AH-r1*I) * x1 = 0, And equation no.2: (A-r2*I) * x2 = 0 time no.1 both sides by x2H ((A*x2)H-r1*x2H)* x1 = 0 Then we have...
  25. U

    Showing A^-1 has eigenvalues reciprocal to A's eigenvalues

    Homework Statement If A is nonsingular, prove that the eigenvalues of A-1 are the reciprocals of the eigenvalues of A. *Use the idea of similar matrices to prove this. Homework Equations det(I\lambda - A) = 0 B = C-1AC (B and A are similar, and thus have the same determinants) The...
  26. S

    Eigenvalues: Matrix corresponding to projection

    Let A be a matrix corresponding to projection in 2 dimensions onto the line generated by a vector v. A) lambda = −1 is an eigenvalue for A B) The vector v is an eigenvector for A corresponding to the eigenvalue lambda = −1. C) lambda = 0 is an eigenvalue for A D) Any vector w perpendicular to...
  27. I

    Can someone please check this work (eigenvalues)

    Homework Statement Let A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right] Find an invertible S and a diagonal D such that S^{-1}AS=D Homework Equations I basically have the question answered, just ONE problem.The Attempt at a Solution My answer is...
  28. P

    Symmetric matrix real eigenvalues

    Homework Statement Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}. Homework Equations The problem says to use the quadratic formula. The Attempt at a Solution From the determinant I get (a-l)(d-l) - b^2 = 0 which...
  29. A

    All eigenvalues 0 implies nilpotent

    Homework Statement How would I go about proving that if a linear operator T\colon V\to V has all eigenvalues equal to 0, then T must be nilpotent? The Attempt at a Solution I know that this follows trivially from the Cayley-Hamilton theorem (the characteristic polynomial is x^n and hence...
  30. L

    The Eigenvalues and eigenvectors of a 2x2 matrix

    Homework Statement Let B = (1 1 / -1 1) That is a 2x2 matrix with (1 1) on the first row and (-1 1) on the second. Homework Equations The Attempt at a Solution A) (1 1 / -1 1)(x / y) = L(x / y) L(x / y) - (1 1 / -1 1) (x / y) = (0 / 0) ({L - 1}...
  31. B

    How Do You Calculate the Probabilities of Measured Values for a Quantum State?

    Homework Statement Suppose that a Hermitian operator A, representing measurable a, has eigenvectors |A1>, |A2>, and |A3> such that A|Ak> = ak|Ak>. The system is at state: |psi> = ((3)^(-1/2))|A1> + 2((3)^(-1/2))|A2> + ((5/3)^(1/2))|A3>. Provide the possible measured values of a and...
  32. R

    Finding eigenvalues with the power series method

    Homework Statement Consider the matrix [1,-5,5;-3,-1,3;1,-2,2] Do four interations of the power method, beginning at [1,1,1] to approximate the dominant eigenvalues of A Homework Equations Matrix multiplication The Attempt at a Solution Okay my issue with this problem is this I...
  33. I

    Eigenvalues of a linear transformation (Matrix)

    Homework Statement Let T: M22 -> M22 be defined by T \[ \left( \begin{array}{cc} a & b \\ c & d \\ \end{array} \right)\] = \[ \left( \begin{array}{cc} 2c & a+c \\ b-2c & d \\ \end{array} \right)\] Find the eigenvectors of T The Attempt at a Solution My...
  34. R

    Eigenvalues of positive definite (p.d) matrix

    If C = A +B where A,B are both p.d, than C is p.d and its eigenvalues are positive. Waht can you say about the relationship between the eigenvalues of C, and A,B ? Thanks.
  35. B

    How to Prove det(xIm - AB) = xm-ndet(xIn - BA)?

    1. The problem statement For integers m >= n, Prove det(xIm - AB) = xm-ndet(xIn - BA) for any x in R. Homework Equations A is an m x n matrix B is an n x m matrix The Attempt at a Solution I tried working out the characteristic polynomials by hand but it just seems too tedious...
  36. P

    Determining Energy Values for a One-Dimension Spin Chain

    Homework Statement Same problem as this old post https://www.physicsforums.com/showthread.php?t=188714 What I'm having problems with is determining the H_{ij} components of the Hamiltonian of a one dimension N site spin chain. And then getting out somehow energy value to prove...
  37. G

    Finding Eigenvalues for 3x3 Matrix

    \left( \begin{array}{ccc} 3 - \lambda & 1 & -1 \\ -4 & 2 - \lambda & 2 \\ -2 & 2 & 2 - \lambda \end{array} \right) (3 - \lambda) \left| \begin{array}{cc} 2 - \lambda & 2 \\ 2 & 2 - \lambda \end{array} \right| + 4 \left| \begin{array}{cc} 1 & -1 \\ 2 & 2 - \lambda \end{array}...
  38. B

    Similar matrices = Same Eigenvalues (NO DETERMINANTS)

    Homework Statement Show that two similar matrices A and B share the same determinants, WITHOUT using determinants 2. The attempt at a solution A previous part of this problem not listed was to show they have the same rank, which I was able to do without determinants. The problem is I...
  39. U

    Does the existence of a ladder operator imply that the eigenvalues are discrete?

    Hi! I don't know much about QM. I'm reading lecture notes at the moment. Angular momentum is discussed. The ladder operators for the angular-momentum z-component are defined, it is shown that <L_z>^2 <= <L^2>, so the z component of angular momentum is bounded by the absolute value of angular...
  40. D

    Normal operators with real eigenvalues are self-adjoint

    Prove that a normal operator with real eigenvalues is self-adjoint Seems like a simple proof, but I can't seem to get it. My attempt: We know that a normal operator can be diagonalized, and has a complete orthonormal set of eigenvectors. Let A be normal. Then A= UDU* for some...
  41. K

    Eigenvalues and one on diagonal matrices 1-

    Hi there, I have some questions to ask about the topic eigenvalues and one on diagonal matrices 1- can a square matrix exist without eignvalues? Do there exists square matrix without eigenvectors corresponding to each of its eignvalues? 2- What is diagonalisation of a matrix, were abouts...
  42. A

    "Proof of Sum of Eigenvalues Inequality

    Homework Statement Proof: \lambda_{\max}(A+B) \leq \lambda_{\max}(A) + \lambda_{\max}(B) Homework Equations Hint from exercise: \lambda_{\max}(A)=\max_{\|x\|=1} x^*Ax The Attempt at a Solution The problem is that the equation on the left side can not be split. So I tried to...
  43. A

    Eigenvalues of a 4x4 matrix and the algebraic multipicities

    Hi everyone Homework Statement Consider the following 4 x 4 matrix: A = [[6,3,-8,-4],[0,10,6,7],[0,0,6,-3],[0,0,0,6]] Find the eigenvalues of the matrix and their multiplicities. Give your answer as a set of pairs: {[lambda1,multiplicity1],[lambda2,multiplicity2],...} 2...
  44. Y

    Must every linear operator have eigenvalues? If so, why?

    It seems to me that http://en.wikipedia.org/wiki/Schur_decomposition" relies on the fact that every linear operator must have at least one eigenvalue...but how do we know this is true? I have yet to find a linear operator without eigenvalues, so I believe every linear operator does have at...
  45. B

    Using Eigenvalues and Eigenvectors to solve Differential Equations

    Homework Statement x1(t) and x2(t) are functions of t which are solutions of the system of differential equations x(dot)1 = 4x1 + 3x2 x(dot)2 = -6x1 - 5x2 Express x1(t) and x2(t) in terms of the exponential function, given that x1(0) = 1 and x2(0) = 0 The Attempt at a Solution I've already...
  46. S

    Eigenvalues and eigenvectors of this matrix

    Consider the nXn matrix A whose elements are given by, A_{ij} = 1 if i=j+1 or i=j-1 or i=1,j=n or i=n,j=1 = 0 otherwise What are the eigenvalues and normalized eigenvectors of A??
  47. T

    Diff EQ Repeated Complex Eigenvalues?

    [b]1. What dimensions of a matrix will give repeated complex Eigenvalues? Give an example of one and show that it has repeated complex Eigenvalues. [b]2. No really equations needed? The Attempt at a Solution My attempt is a 2x2 which i don't think is right but here it is. If...
  48. A

    Simultaneous diagonalization and replacement of operators with eigenvalues ?

    Apparently, if I have a Hamiltonian that contains an operator, and that operator commutes with the Hamiltonian, not only can we "simultaneously diagonalize" the Hamiltonian and the operator, but I can go through the Hamiltonian and replace the operator with its eigenvalue everywhere I see it...
  49. D

    Understanding Eigenvalues in Rotational Transformations: A False Assertion

    Homework Statement True/False If Ttheta is a rotation of the Euclidean plane R2 counterclockwise through an angle theta, then T can be represented by an orthogonal matrix P whose eigenvalues are lambda1 = 1 and lambda2 = -1. Homework Equations The Attempt at a Solution Just checking to see...
  50. D

    Why Am I Getting Only Two Eigenvalues for This Matrix?

    Homework Statement Find the eigenvalues of the following matrix: \left( \begin{array}{ccc} 1 & 0 & -3 \\ 1 & 2 & 1 \\ -3 & 0 & 1 \end{array} \right) Homework Equations The Attempt at a Solution I think I'm forgetting a basic algebra rule or something. I know there are supposed to be 3...
Back
Top