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. Roodles01

    Hamiltonian matrix and eigenvalues

    OK. An example I have has me stumped temporarily. I'm tired. General spin matrix can be written as Sn(hat) = hbar/2 [cosθ e-i∅sinθ] ...... [[ei∅sinθ cosθ] giving 2 eigenvectors (note these are column matrices) I up arrow > = [cos (θ/2)] .....[ei∅sin(θ/2)] Idown arrow> =...
  2. hilbert2

    Can small changes in fundamental constants affect the properties of water?

    Suppose we have a matrix A that has eigenvalues λ1, λ2, λ3,... Matrix B is a matrix that has "very small" matrix elements. Then we could expect that the eigenvalues of sum matrix A + B would be very close to the eigenvalues λi. But this is not the case. The eigenvalues of a matrix are not...
  3. W

    Can we find Eigenvalues for simultaneous equation?

    hi, please tell me what are the limitations for finding eigenvalues ? thanks
  4. O

    Sketching Graphs using Eigenvalues

    Homework Statement For the conic, 5x2+4xy+5y2=9, find the direction of the principal axes, sketch the curve. I found the eigenvalues as 3,7 but have no idea whether the 'new' equation is 3(x')2+7(y')2 or 7(x')2+3(y')2 is there a way to determine which 'way' it goes? I took a guess...
  5. M

    Fourier Series/transform and eigenvalues

    Hello Physics Forums community, I'm afraid I really need a hand in understanding Why are the Fourier Series for continuous and periodic signals using diferent notation of the Fourier Series for discrete and periodic Signals. I have been following the book " Signals and Systems " by Alan V...
  6. nomadreid

    Common eigenvalues for two states or two bases of same state?

    Two questions: If you have two states which have at least one common eigenvalue, then are the two states distinguishable? If you have one state but measure it with two different bases, can one conclude anything if the two measurements have a common eigenvalue? Thanks
  7. K

    Three masses two strings system: lagrange and eigenvalues

    Homework Statement We have a three mass two strings system with: m_1 string M string m_2 The end masses are not attached to anything but the springs, the system is at rest, and k is equal for both strings and m_1 and m_2 are equal. The distance between to m_1 and m_2, on both sides of M...
  8. M

    Powers of a Matrix and Eigenvalues proof

    Homework Statement Prove that if A is an nxn matrix with eigenvector v, then v is an eigenvector for Ak where kε(all positive integers) Homework Equations Av=λv The Attempt at a Solution Av=λv A(Av)=A(λv) Akv=λ(Av) i know i may not be doing it right but this is what i can...
  9. M

    Similar Eigenvalues of Invertible Matrices

    Homework Statement Let A and C be nxn matrices with C invertible. Prove that A and C-1AC have the same eigenvalues. Homework Equations B=C-1AC The Attempt at a Solution det(A-λI) =det(B-λI) det(A-λI) =det(C-1AC - λI) det(A-λI) =det(C-1AC - λC-1IC) det(A-λI) =det[CC-1(A-λI)]...
  10. B

    Role of eigenvalues in phase portraits

    Hi, In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard examples like prey-predator, pendulum etc. I have a rather unclear thought in my head regarding the role of real/imaginary eigenvalues in the system...
  11. Fernando Revilla

    MHB Null and non null eigenvalues (Oinker's question at Yahoo Answers)

    Here is the question: Here is a link to the question: Matrix Question?? a.b.c.d.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  12. P

    Eigenvalues, eigenvectors, eigenstates and operators

    Homework Statement Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could...
  13. P

    An eigenstates, eigenvectors and eigenvalues question

    Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right...
  14. L

    Find the basis for both eigenvalues

    Homework Statement Given matrix A= {[39/25,48/25],[48/25,11/25]} find the basis for both eigenvalues. Homework Equations The Attempt at a Solution I row reduced the matrix and found both eigenvalues. I found λ = -1, and λ = 3. Then, I used diagonalization method [-1I2 - A 0]...
  15. D

    Geometric Multiplicity of Eigenvalues

    Could someone please explain to me (with an example if possible) what is the Geometric Multiplicity of Eigenvalues? I cannot understand it from what I have read on the web till now. Thanks in advance.
  16. P

    Understanding Eigenvalues and Eigenvectors: A Beginner's Guide

    can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?
  17. F

    Doubt about exercise with eigenvalues

    Homework Statement Given the endomorphism ϕ in ##\mathbb{E}^4## such that: ϕ(x,y,z,t)=(4x-3z+3t, 4y-3x-3t,-z+t,z-t) find: A)ker(ϕ) B)Im(ϕ) C)eigenvalues and multiplicities D)eigenspaces E)is ϕ self-adjoint or not? explain The Attempt at a Solution I get the associated matrix...
  18. F

    Linear algebra: eigenvalues, kernel

    Homework Statement I've tried to solve the following exercise, but I don't have the solutions and I'm a bit uncertain about result. Could someone please tell if it's correct? Given the endomorphism ##\phi## in ##\mathbb{E}^4## such that: ##\phi(x,y,z,t)=(x+y+t,x+2y,z,x+z+2t)## find: A) ##...
  19. S

    Eigenvalues and Eigenvectors of a 2x2 Matrix P

    Homework Statement Find the eigenvalues and eigenvectors of P = {(0.8 0.6), (0.2 0.4)}. Express {(1), (0)} and {(0), (1)} as sums of eigenvectors. Homework Equations Row ops and det(P - λI) = 0. The Attempt at a Solution I've found the eigenvectors and eigenvalues of P to be 1...
  20. B

    Find the eigenvalues of a given matrix

    [b]1. The 3x3 Matrix A=[33, -12, -70; 0, 1, 0; 14, -6, -30] has three distinct eigenvalues, λ1<λ2<λ3. Find each eigenvalue.[b]2. det(A-λI)=0 where I denotes the appropriate identity matrix (3x3 in this case)[b]3. Here's my attempt: --> det([33, -12, -70; 0, 1, 0; 14, -6, -30]-λ[1, 0, 0; 0, 1...
  21. I

    Extracting eigenvalues from wavefunction

    Homework Statement The Hamiltonian for a rigid rotator which is confined to rotatei n the xy plane is \begin{equation} H=-\frac{\hbar}{2I}\frac{\delta^{2}}{\delta\phi^{2}} \end{equation} where the angle $\phi$ specifies the orientation of the body and $I$ is the moment of inertia...
  22. B

    Eigenvalues of unitary operators

    Homework Statement We only briefly mentioned this in class and now its on our problem set... Show that all eigenvalues i of a Unitary operator are pure phases. Suppose M is a Hermitian operator. Show that e^iM is a Unitary operator. Homework Equations The Attempt at a Solution...
  23. W

    PDE's Find the values of lambda (eigenvalues)

    the problem stays to find the values of Lambda for which the given problem has nontrivial solutions. Also to determine the corresponding nontrivial eigenfunctions. y''-2y'+\lambday=0 0<x<\pi, y(0)=0, y(\pi)=0 r^{2}-2r=-\lambda r=1±i\sqrt{\lambda+1}...
  24. N

    Eigenvalues of nonlinearly coupled equations

    Hi everyone, I am currently dealing with a nonlinear system of coupled equations. In fact I had performed a perturbation approach for this system which is highly nonlinear. Thanks to first step of the perturbative approach I could reach eigenvalues in the "linear case". Right now I want to...
  25. H

    Finding eigenvalues of a 3x3 matrix

    Homework Statement Find the eigenvalues | 1 2 -1| | -5 7 -5 | | -9 8 -7| Homework Equations The Attempt at a Solution I know that i need to add a -λ to every term in the trace so my matrix becomes | 1-λ 2 -1| | -5 7-λ -5| | -9 8 -7-λ| Then i need to...
  26. B

    Eigenvalues of a rank 1 matrix?

    How come a square matrix has eigenvalues of 0 and the trace of the matrix? Is there any other proof other than just solving det(A-λI)=0?
  27. S

    Conic Formula Eigenvalues and PDEs

    Homework Statement We have the following conic formula ##ax^2 + 2bxy + cy^2 + dx + ey = ## constant which corresponds to a ellipse, hyperbola or parabola. The second order terms of the corresponding PDE $$ a\frac{\partial^2 u}{\partial x_1^2} + 2b\frac{\partial^2 u}{\partial x_1\partial x_2} +...
  28. fluidistic

    Eigenvalues of the position operator

    I'm new to QM, but I've had a linear algebra course before. However I've never dealt with vector spaces having infinite dimension (as far as I remember). My QM professor said "the eigenvalues of the position operator don't exist". I've googled "eigenvalues of position operator", checked into...
  29. R

    Linear system of differential equations with repeated eigenvalues

    Homework Statement X'=AXA=\left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 &0 \\0 & 0 & -1\end{matrix}\right] Homework Equations n/a The Attempt at a Solution The eigenvalues are -1, and \pm i. I also can see that the matrix A is already in the form A=\left[\begin{matrix} \alpha & \beta & 0 \\...
  30. B

    Finding eigenstates and eigenvalues of hamiltonian

    Hey there, the question I'm working on is written below:- Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by: H = |a'>∂<a''| + |a''>∂<a'| where ∂ is just a real number. Write down the eigenstates...
  31. G

    Is it possible to have a diagonal matrix with all eigenvalues = zero ?

    Homework Statement If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors? Homework Equations The Attempt at a Solution
  32. D

    Eigenvalues of Matrix Function

    Homework Statement Define a matrix function f(T) of an nxn matrix T by its Taylor series f(T)=f0 +f1T +f2T2+... Show that if matrix T has the eigenvalues t1,t2...tn, then f(T) has eigenvalues f(t1), f(t2)...f(tn) Homework Equations The Attempt at a Solution I am at a loss of how...
  33. B

    Collapse of state vector for continuous eigenvalues

    1. In the many statements of the QM postulates that I've seen, it says that if you measure an observable (such as position) with a continuous spectrum of eigenvalues, on a state such as then the result will be one of the eigenvalues x, and the state vector will collapse to the...
  34. Q

    Eigenvalues of 12*12 symbolic matrix

    Hi dear friends I have a 12*12 symbolic matrix in terms of x y z d that I want its eigenvalues but not mathematica nor MATLAB can do it for me.My mathematica is "7" so If you have a newer version or even in MATLAB , would you mind checking my matrix in your software? this is my matrix in...
  35. N

    Vibration Analysis, Eigenvalues

    Hello, I have a 3D COMSOL model which I am using for the purpose of vibration analysis. Up to now I've got analytical eigenvalues using COMSOL with MATLAB. I have to correlate the results with the measured eigenvalues results, using MAC criterion. The problem is that my analytical results...
  36. Fernando Revilla

    MHB Sarah Morash's question at Yahoo Answers about eigenvalues

    Here is the question: Here is a link to the question: Help finding the eigenvalues of a matrix? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  37. schmiggy

    Complex eigenvalues - solve the system

    Homework Statement Using eigenvalues and eigenvectors, find the general solution to dx/dt = x - y dy/dt = x + yHomework Equations Matrix 'A' - lambda*identity matrix ; for finding eigenvalues and thus eigenvectors Other relevant equations written on the attached scanned image of my attempt at...
  38. T

    Physical Chemistry/Quantum Mechanics Eigenvalues

    Homework Statement Indicate which of the following expressions yield eigenvalue equations and identify the eigenvalue. a) d/dx (sin(∏x/2)) b) -i*hbar * ∂/∂x (sin(∏x/2)) c) ∂/∂x (e-x^2) The Attempt at a Solution I know that if the wave equation yields an eigenvalue equation, it will...
  39. Y

    MHB Technical problem with eigenvalues

    Hello I was trying to find eigenvalues of a matrix. I calculated the characteristic polynomial by calculating (A-lambdaI) and then calculating it's determinant. The results was: -\lambda ^{3}+8\lambda ^{2}-20\lambda +16 which is the correct calculation. Now, the eigenvalues are 2,2,4, but I...
  40. O

    MHB Significance of the Eigenvalues of a covariance matrix

    Hello everyone! I'm curious to know what is the significance of the Eigenvalues of a covariance matrix. I'm not interested to find an answer in terms of PCA (as you of you may be familiar with the term). I'm thinking of a Gaussian vector, whose variance represent some notion of power or...
  41. F

    How do I solve the eigenvalues equation for a 3x3 matrix?

    Homework Statement Find the eigenvalues of the following and the eigenvelctor which corresponds to the smallest eigenvalue Homework Equations I know how to find the eigenvalues and eigenvectors of a 2x2 matric but this one I'm not so sure so any help would be appreciated The...
  42. matqkks

    MHB Repeated eigenvalues of a symmetric matrix

    I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?
  43. matqkks

    Repeated eigenvalues of a symmetric matrix

    I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?
  44. S

    Eigenvalues of a compact positive definite operator

    eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...
  45. S

    Eigenvalues of a complex symmetric matrix

    Eigen values of a complex symmetric matrix which is NOT a hermitian are not always real. I want to formulate conditions for which eigen values of a complex symmetric matrix (which is not hermitian) are real.
  46. A

    Why most observables have real eigenvalues

    I have always been quite confused about the fact that any measurement MUST yield a real number. What says it must so? Don't we modify our measurement apparatus to yield something which is consistent with the theory. So coulnd't we just imagine having complex values for momentum and position. All...
  47. L

    ODE Linear System Complex Eigenvalues

    Homework Statement Solve the following systems by either substitution or elimination: dx/dt = y dy/dt = -x + cos(2t) Homework Equations I know the solution is: x(t) = c_1cos(t) + c_2sin(t) - 1/3cos(2t) y(t) = -c_1sin(t) + c_2cos(t) + 2/3sin(2t) The Attempt at a Solution x' = [ 0 1; -1...
  48. S

    Proving that the eigenvalues of a Hermitian matrix is real

    Homework Statement Prove that the eigenvalues of a Hermitian matrix is real. http://www.proofwiki.org/wiki/Hermitian_Matrix_has_Real_Eigenvalues The website says that "By Product with Conjugate Transpose Matrix is Hermitian, v*v is Hermitian. " where v* is the conjugate transpose of v...
  49. D

    MHB Finding Eigenvalues for Different r Values

    $$ \mathcal{J} = \begin{pmatrix} -\sigma & \sigma & 0\\ 1 & -1 & -\sqrt{b(r - 1)}\\ \sqrt{b(r - 1)} & \sqrt{b(r - 1)} & - b \end{pmatrix} $$ From a quick try and error, I was able to find that when $r = 1.3456171$ we will have 3 negative eigenvalues. But when $r = 1.3456172$, there will be a...
  50. S

    Negative energy eigenvalues of Hamiltonian

    Homework Statement If I have a Hamiltonian matrix, \mathcal{H}, that only depends on a kinetic energy operator, do the energy eigenvalues have to be non-negative? I have an \mathcal{H} like this, and some of its eigenvalues are negative, so I was wondering if they have any physical...
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