Eigenvalue Definition and 402 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.

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  1. A

    Operators satisfying abstract commutation relation; then finding an eigenvalue.

    So, my problem statement is: Suppose that two operators P and Q satisfy the commutation relation [Q,P] = Q . Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. This shouldn't be too difficult, but...
  2. C

    Hamiltonian of the Quantum Harmonic Oscillator-Eigenfunction & Eigenvalue

    Homework Statement Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue. Homework Equations u1(q)=A*q*exp((-q^{2})/2) The Attempt at a Solution Ok, so I know that the Quantum Harmonic Oscillator...
  3. B

    Eigenvalue VS Cholesky Decomposition

    Assuming the matrix is positive definite (necessary for cholesky decomposition). Which is faster? Which is more accurate? Is there a reliable source that has all the most common decompositions listed in order of accuracy and speed?
  4. B

    Fortran eigenvalue decomposition

    I've been trying to invert a real symmetric matrix and the inverse that I compute via eigenvalue decomposition is not the inverse (using QV^-1Q^T), the stranger thing is that QVQ^T gets back my orginal matrix matrix. Even more unusual is that the matrix starts off at approximately identity (in...
  5. A

    Eigenvalue Question (p2.14 in Quantum Chemistry and Spectroscopy)

    function: e^-(x^2/2) operator: d^2/dx^2 -x^2 The answer key says the function is an eigenfunction of the operator with an eigenvalue of -6. I can't figure out how to reach this conclusion. Also, Wolfram Alpha says d/dx(d/(dx)e^(-x^2/2)) = e^(-x^2/2) (x^2-1). Isn't this inconsistent with...
  6. C

    What are the eigenvalues of P(A)?

    Homework Statement αo, α1,..., αd \inℝ. Show that αo + α1λ + α2λ2 + ... + αdλd \inℝ is an eigenvalue of αoI + α1A + α2A2 + ... + αdAd \inℝ^{nxn}. 2. The attempt at a solution If λ is an eigenvalue of A, then |A - Iλ| = 0. Also, λn is an eigenvalue An. So we basically have to somehow...
  7. C

    Homework SolutionEigenvalue of a Matrix: Proof Involving Nonsingular Matrices

    Proof involving nonsingular matrices. Homework Statement If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix. I've proved it like this: Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B. B-1 = (I - A)-1(I +...
  8. I

    Eigenvector eigenvalue proof problem

    Homework Statement Let A and B be symmetric matrices and X is a vector in the eigenvalue problem AX-λBX=0 a) Show that the eigenvectors are orthogonal relative to A and B. b) If the eigenvectors are orthonormal relative to B , determine C such that (C-λI)X=0, where C is a diagonal...
  9. D

    Physical meaning of zero eigenvalue

    Hello, Given the hamiltonian : H = -( aS_z^2 + b(S_+^2 +S_-^2) ) with S=1 and a,b>0 are constants. working with the base: { |m=1> , |m=-1> , |m=0> } The matrix form of H is: H = \left( \begin{array}{ccc} -ah^2 & -bh^2 & 0 \\ -bh^2 & -ah^2 & 0 \\ 0 & 0 & 0 \end{array}...
  10. W

    Can Non-Homogenous Simultaneous Equations be Solved using Eigenvalues?

    hi i know how to calculate eigenvalue of given matrix. I want to know if two non homogenous simutaneous equation are given - than can we find its eigenvalue.
  11. I

    Linear algebra: eigenvalue & character polynomials proof

    we are given B = CAC^-1 Prove that A and B have the same characteristic polynomial given a hint: explain why ƛIn = CƛInC^-1 what I did was: B = CAC^-1 BC = CA Det(BC) = Det(CA) Det(B) Det(C) = Det(C) Det(A) Now they’re just numbers so I divide both sides by Det(C) Det(B) = Det(A)...
  12. 3

    Eigenvalue for Orthogonal Matrix

    Homework Statement Let Q be an orthogonal matrix with an eigenvalue λ_{1} = 1 and let x be an eigenvector belonging to λ_{1}. Show that x is also an eigenvector of Q^{T}. Homework Equations Qx = λx where x \neq 0 The Attempt at a Solution Qx_{1} = x_{1} for some vector x_{1}...
  13. 3

    Do Rotational Matrices Always Yield Real Eigenvalues?

    Homework Statement Show that the matrix A = [cos θ -sin θ sin θ cos θ] will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result. Homework Equations Ax = λx, so det(A-λI) = 0 The Attempt at a Solution In this case...
  14. P

    Understanding the Significance of Eigenvalues in Quantum Mechanics and Physics

    So I understand the idea of eigenvalues, eigenvectors, and eigenfunctions corresponding to a given operator on some vector or function space. But I'm just wondering, why are eigenvalues so important in quantum mechanics and physics in general? What I mean is, why are scaled multiples of a...
  15. Y

    How to Solve a Hamiltonian with a Complex Term?

    Homework Statement I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} Homework Equations The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The...
  16. T

    Geometric multiplicity of an eigenvalue

    Say we have an eigenvalue \lambda and corresponding eigenvectors of the form (x,x,2x)^T. What is the geometric multiplicity?
  17. C

    What is the eigenvalue problem for the given matrix and how can it be solved?

    Homework Statement The problem amounts to finding the eigenvalues of the matrix |0 1 0| |0 0 1| |1 0 0| (I have no idea how to set up a matrix in the latex format, if anyone can tell me that'd be great) Homework Equations The characteristic equation for this matrix is...
  18. M

    Eigenvalue solutions of the transcendental equation

    Homework Statement Given the Sturm-Liouville system: y'' + λy = 0 , y(0) - y'(0) = 0 , y(1) + y'(1) = 0 Show using the Rayleigh Quotient that the eigenvalues are positive. Show that these eigenvalues are given as the solutions of the transcendental equation: tan ( √λ ) =...
  19. L

    MATLAB Finding the Smallest Eigenvalue with Power Iteration in MatLab

    I'm tinkering with a code snippet where a part finds eigenvalues. eig(A); The thing is, I tried to do it not using eig() to grasp this and got stuck. Could anyone shed some light on this..? How do I find the smallest eigenvalue?
  20. R

    Eigenvalue eqn for a electron in a one-dimensional lattice

    Homework Statement An electron moves in a one-dimensional lattice with the separation between adjacent atoms being equal to a. a. Write down the momentum eigenvalue equation for the electron. b. Find the general form of the solutions of the eigenvalue equation. c. By requiring that the...
  21. N

    Eigenvalue Proof: Proving A^2=A has 0 or 1 as an eigenvalue

    Homework Statement Proof: Prove that if A is an nxn (square mtx) such that A^2=A, then A has 0 or 1 as an eigenvalue. The Attempt at a Solution A=A^2 A^2-A=0 A(A-I)=0 A=0 or A=1 and then plugging the A solutions into the characteristic equation and solving for λ
  22. S

    Eigenvector for Complex Eigenvalue help

    In my lecture notes my prof used the eigenvalue c= 1 + i and ended up with the matrix with (5 3+i) as row 1, and the second row is zeroes. After that, he simply wrote that the basis for this eigenvalue c is (3+i,-5) (in column form) without explaining. How did he get that basis? I tried working...
  23. Hootenanny

    Mathematica Generalised Eigenvalue Problem in Mathematica

    I have a generalised eigenvalue problem of the form A\boldsymbol{u} = \lambda B\boldsymbol{u}\;, where A and B are symmetric matrices with real symbolic entries. I'm trying to compute the eigenvalues with Mathematica using the command Eigenvalues[{A,B}] which according to the documentation...
  24. A

    Direct Eigenvalue solver, which retains the the order of eigenvectors

    Hi guys I have a problem that I need some help with, I am looking for a direct eigenvalue solver algorithm. The problem is that all the eigenvalue solvers I can find seems to reorder the final matrix after the size of the eigenvalues. The matrix it shall calculate is very small (5-10)...
  25. M

    What Is the Multilinear Eigenvalue Problem Called?

    I'm trying to do something that requires solving an eigenvalue problem of the form A_{imkl} c_m c_k c^*_l=\lambda c_i where A is a known rank-4 tensor, \lambda is the eigenvalue, and the c_i's are a set of unknown coefficients that I need to determine. I would guess that this type of problem...
  26. A

    True or false eigenvalue problem

    Q)Different eigenvectors corresponding to an eigenvalue of a matrix must be linearly dependant? Is the above statement true or false.Give reasons.
  27. P

    Solving for complex phase speed eigenvalue

    Hey there, first timer poster here Homework Statement I'm working on a barotropic linear instability analysis and I've been having trouble getting an expression for the complex phase speed eigenvalue C = C_r + i*C_i for the purpose of plotting a dispersion diagram (C_i vs k or C_r vs...
  28. I

    Conditions for Unique Eigenvalues and Solving Systems with Diagonal Matrices"

    Homework Statement Consider the matrix A=[a d f; 0 b e; 0 0 c], where all elements are real numbers (a) what condition(s) on the elements of A are sufficient to guarantee that A has 3 distinct eigenvalues? (b) prove that any two eigenvectors x1 and x2 associated with two distinct...
  29. N

    Solve Eigenvalue Problem A: Proving (λI-A)=0 with Simetric Matrices

    A is a simetric metrices nxn. so v\in R^n and v\neq 0 so (\lambda I -A)^2=0 for some \lambda prove that for the same v (\lambda I -A)=0 how i tried to solve it: i just collected data from the given. simetric matrices is diagonizable. B=(\lambda I -A) we were given that B^2v=0 so...
  30. T

    Why is the probability of measuring an eigenvalue its coefficient squared?

    Homework Statement This is an example from Gasiorowicz's Quantum Physics. "Example 3-1" is a particle in an infinite potential-well, but that should not matter. Homework Equations The Attempt at a Solution Why is P(-2) (which I suppose is the probability that the eigenvalue -2 is...
  31. E

    SVD vs Eigenvalue Decompositon (Diagonalizability)

    Okay, I know that if I can't get n linearly independent eigenvectors out of a matrix A (∈ℝnxn), it is not diagonalizable (and that some necessary conditions for diagonalizability in this regard may be being symmetric and/or having distinct eigenvalues.) This is how things are for the usual...
  32. D

    Phonon Modes and Vanishing Eigenvalues in Periodic Lattices

    Dear all, in these http://pages.unibas.ch/comphys/comphys/TEACH/SS04/course.pdf" lecture notes, the author says on page (0-120): http://img15.imageshack.us/img15/615/capturena.png It is not obvious to me, why due to the translation invariance of the energy 3 eigenvalues of the D_IJ matrix have...
  33. S

    What is the Eigenvalue Equation for a 2D Harmonic Oscillator?

    Homework Statement Please take a look at the attachment for the problem statement. Homework Equations For 1 dim Harmonic oscillator, E = (n+1/2)h.omega/2pi I don't know which equation to use for 2 dim The Attempt at a Solution I am unable to solve because I don't know which...
  34. I

    Find 2 eigenvectors given an eigenvalue, and find remaining eigenvalue

    Could someone please walk me through answering this question: it looks easy but i forgot how to do it -__- i did A - lamda I = 0, substituted lamda = 5, then my matrix became 0 4 4 0 -2 -2 0 -2 -2 so i put in the form of A|b where b was a zero vector and i row reduced, getting 0 1 1 | 0 0...
  35. D

    Linear algebra Eigenvalue related

    Homework Statement a) |-1 1 1| | 1 -1 1| = A | 1 1 -1| Find an orthoginal matrix P that diagonalizes Ab) |0 1| What value of a is multiplicity 2, what value of a is eigen values -1 and 2 A = |a 1| what value of a does A have real eigenvaluesC) If A is a...
  36. O

    Finding the lowest eigenvalue, Rayleigh-Ritz method, Calculus

    I have a Sturm-Liouville system \frac{d}{dx}p(x)\frac{du}{dx} - q(x)u(x)+\lambda \rho(x) u(x) = 0 with p(x) = (1-x^2)^{2p} q(x) = k^2 \rho(x) = (1-x^2)^{p-1} (p,k^2 are positive real) u(x) is defined on the interval (-A,A) where 0<A<=1. Boundary condition that u(x) satisfies is...
  37. U

    Does this eigenvalue problem has degenerate solutions?

    Hello everyone, I am solving an eigenvalue problem. Right now, I would like to know; How to determine the degeneracy of eigensolution of sturm-liouville differential eigenvalue problem? I have an eigenvalue sturm-liouville problem H f(y) = E f(y) where H is a differential operator and E is...
  38. WannabeNewton

    Solving Eigenvalue Problems: Choosing Eigenvectors

    I have a rather basic question about solving eigenvalue problems. Once you actually find all the eigenvalues for a given operator in some basis and you go about finding the respective eigenvectors through the components and run into a situation like this: \mid \omega = 1 > \Rightarrow...
  39. M

    Can Any LTI System Be Characterized by Its Impulse Response or Eigenvalues?

    Hello everyone, please help me to answer this question. Is this true that any LTI system can be characterized by either its impulse response or engenvalue?
  40. M

    Is k an Eigenvalue of A with Sum of Row Entries as k?

    1. Homework Statement Suppose that A is a square matrix and the sum of the entries of each row is some number k. Is k an eigenvalue of A? if so, what is the corresponding Eigenvector?2. Homework Equations Ax-λx=0 3. The Attempt at a Solution (1-k)(K-λ)-k=0I am not sure how to solve this...
  41. C

    Eigenvalue method for homogeneous eq's

    I am working on a problem and before I post the remaining questions on it, I want to be sure I calculated the eigenvector correctly. The eigenvalue I used was lambda = 3-4i. \begin{bmatrix} 3-lambda & -4\\ 4 & 3-lambda\end{bmatrix} After substituting, the eigenvector I came up with is V...
  42. M

    Linear algebra. invertible matrix and its eigenvalue

    Homework Statement Let A be an invertible matrix. show that if λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^-1 PLEASE HELP ME . Thank you.
  43. C

    Quantum, finding energy eigenvalue spectrum

    Homework Statement The question says for the hamiltonian \hat{}H+\hat{}H1 calculate the complete energy eigenvalue spectrum. for the ground state show that the result agrees with the one found by the perturbation theory previously. I'd assume \hat{}H here is just the standard...
  44. D

    Eigenvalue of x^4 Potential: A Mystery Explored

    So I'm working out on a potentials of the type x^2p, and I have a program that solves and gives the eigenenergies for a potential that I have (x^n in general). I noticed that for a ground state the potential x^4 has the smallest eigenvalue : 0.667981 in units where \hbar=m=\omega=1. I...
  45. B

    Master the Eigenvalue Algorithm for Math GRE Exams

    I'm taking the math subject GRE in just over a year's time... and I was wondering if there are "ideal" algorithms to have in our tool box to do a computation like this quickly. Obviously the type of matrices in a standardized exam are going to be fairly clean or look dirty but have some less...
  46. X

    Understanding Eigenvalue Problems

    We are doing Eigenvalue problems in my Differential Equations class and I just want to make sure I understand some of these concepts. If anyone could look through my current understanding and guide me in the right direction that would be great! So when you have some equation L[y]=\lambda...
  47. T

    Determine unit normal (eigenvalue, eigenvector)

    Homework Statement For a material the stress is defined by the means of the stress matrix O O = (6 1 -2 1 2 2 -2 2 5) Expressed in MPA It can be derived that the principe stress are: O1= 4-sqrt(13), O2= 5 and O3=4+sqrt(13) I know you can derive the principal...
  48. jegues

    Finding Eigenvalues and Eigenfunctions for O.D.E. Problem on Interval [0, 4π]

    Homework Statement Solve the differential equation eigenvalue problem: f'' + \lambda f = 0, \quad 0 \leq x \leq 4\pi, \quad \text{where} \quad f^{'}(0) =0, \quad f^{'}(4\pi) = 0, \quad \text{and} f \neq 0. Consider ONLY \quad \lambda \geq 0, \quad and find the values of \quad \lambda...
  49. B

    Power of matrix and power of eigenvalue

    Assuming that k\geq0, How does one prove that when A has an eigenvaule \lambda that A^{k} has an eigenvalue \lambda^{k}?
  50. 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
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