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. T

    Proving that the matrix is invertible given the eigenvalues?

    Homework Statement Given an unknown matrix with eigenvalues 1,2,3, prove that it is invertible? The Attempt at a Solution If the det = 0, then there exists an eigenvalue = 0. Since none of the eigenvalues are 0, then the det ≠ 0 and thus the matrix is invertible. Is this a valid proof?
  2. L

    Finding Eigenvalues and determine if there are invariant lines

    Homework Statement Find the eigen values of the following mapping and determine if there are invariant lines. (2 -4) (-3 3) is the mapping. Homework Equations det (L-λI)=0 The Attempt at a Solution L-λI= (2-λ -4) (-3 3-λ) det(L-λI)=0=ac-bd=(3-λ)(2-λ)-12 ...
  3. J

    Adding a constant times the unit matrix and eigenvalues

    To find the eigenvalues \lambda of a matrix A you solve the equation det |A - \lambda I| = 0 eq(1) but now what if you add e I to the matrix A where e is a constant? Then you have to solve the equation, det |(A + eI) - \lambda_{new} I| = 0 eq(2) which is the same as solving...
  4. R

    Linear Algebra - Matrix with given eigenvalues

    Homework Statement Come up with a 2 x 2 matrix with 2 and 1 as the eigenvalues. All the entries must be positive. Then, find a 3 x 3 matrix with 1, 2, 3 as eigenvalues. The Attempt at a Solution I found the characteristic equation for the 2x2 would be λ2 - 3λ + 2 = 0. But then I couldn't get...
  5. C

    Are There Infinite Eigenfunctions with Distinct Eigenvalues for y''+λy=0?

    Homework Statement Show that y''+\lambda y=0 with the initial conditions y(0)=y(\pi)+y'(\pi)=0 has an infinite sequence of eigenfunctions with distinct eigenvalues. Identify the eigenvalues explicitly.Homework Equations The Attempt at a Solution \lambda \le 0 seems to yield the trivial...
  6. M

    Solving homogeneous system involving decimal eigenvalues

    Homework Statement I need to find the general solution of the system [3 5] [-1 -2] Homework Equations so to get the eigenvalues, det(A - λI) The Attempt at a Solution determinant is (3-λ)(-2-λ) + 5 which would be λ2 - λ - 1 so by the quadratic equation the eigenvalues are...
  7. V

    Eigenvalues of Matrix Addition

    Hello, I have a linear algebra problem that I need help with. Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large matrices (6^n x 6^n, where n>= 3, to be specific). Currently, we are just using MATLAB's eig() function to get them. I...
  8. E

    LQR Design: Choosing Q & R matrices for specific eigenvalues

    Assuming we have a closed loop system (A-BK), with stable eigenvalues, how would one choose matrices Q and R such that the eigenvalues of (A-BK) are exactly [-1,-2]? LTI System: \dot{x}=\left[ \begin{array}{cc} 0& 1 \\ 0 & 0 \\ \end{array} \right]x+\left[ \begin{array}{c} 0 \\ 1 \\...
  9. H

    Eigenvalues of a tridiagonal matrix

    Some algorithms of finding the eigenvalues of symmetric matrices first transform the matrix to a tridiagonal matrix which is similar to the original matrix and then find the eigenvalues of the tridiagonal matrix. . Are there special algorithms for a tridiagonal matrix, or do the same algorithms...
  10. R

    Bifurcations and Chaos - Complex eigenvalues

    Homework Statement Identify the stable, unstable and center eigenspaces for \dot{y} = the 3x3 matrix row 1: 0, -3, 0 row 2: 3, 0, 0 row 3: 0, 0, 1 Homework Equations The Attempt at a Solution This is an example used from the lecture and I understand how to get the...
  11. Totalderiv

    Solving Complex Eigenvalues Homework

    Homework Statement Apply the eigenvalue method to find a general solution of the given system. x_1' = 5x_1 - 9x_2 x_2' = 2x_1 - x_2 Homework Equations (A-λI)v=0 The Attempt at a Solution x_1' = 5x_1 - 9x_2 x_2' = 2x_1 - x_2 \left[ \begin{array}{cc} 5-λ & -9\\ 2 & -1-λ...
  12. P

    PDE question: Eigenvalues

    Homework Statement Let λ_n denote the nth eigenvalue for the problem: -Δu = λu in A, u=0 on ∂A (*) which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions. (i) Show that (*) has no...
  13. D

    Finding Eigenvalues and C1 & C2

    Homework Statement (attached)Homework Equations The Attempt at a Solution I really don't know where to start. There is nothing given for me to start with. And the instruction says "Choose" so am I really suppose to really choose or do you guys any idea how to start this? *I know that...
  14. G

    Eigenvalues of 4x4 Hermitian Matrix (Observable)

    Homework Statement Find the allowed energies for a spin-3/2 particle with the given Hamiltonian: \hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z} The Attempt at a Solution The final matrix I get is: \begin{pmatrix} \frac{3}{2} & 0 &...
  15. C

    Matrices, Proof and Eigenvalues.

    Homework Statement Looking for some help with the proof if possible. Vector r = x y z Rotation R = cos(θ) 0 sin(θ) 0 1 0 -sin(θ) 0 cos(θ) r' = Rr It asks me to prove that r'.r' = r.r Second part of the question is about eigenvalues, it asks me to find the three...
  16. H

    Eigenvalues of the product of two matrices

    Hello everyone, Before I ask my question, be informed that I haven't had any formal course in linear algebra, so please forgive me if the question has a well-known answer. I have two symmetric matrices, A and B. We know the eigenvalues and eigenvectors of A, and B. Now I need to...
  17. D

    Eigenvalues of coefficient matrix problem

    Homework Statement Determine the general solution of the system of homogeneous differential equations. The system of homogeneous differential equations is: X'_{1}(t) = 141x_{1}(t) - 44x_{2}(t) X'_{2}(t) = 468x_{1}(t) - 146_{2}(t) What is Eigenvalues of Coefficient Matrix? What is...
  18. J

    Numerical approximation of the eigenvalues and the eigenvector

    Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...
  19. S

    What is the Correct Way to Find Eigenvalues and Eigenvectors of a Matrix?

    Find the eigenvalues and corresponding eigenvector of the matrix. A= [-4 4 8 ] [0 0 -10] [0 0 2 ] [1 -1 0] ~ [0 0 1 ] [0 0 0 ] I calculated by A = -\lambdaI So, [1-lamda -1 0 ] [0 -lamda 1] [0 0 -lamda] so, lamda = 0,0, and 1 So I got...
  20. N

    Find the equilibrium solution and eigenvalues and eigenvectors of system?

    Hey guys, I need to find the equilibrium solution (critical point) for the given system. Also I need to take the homogeneous equation x' = Ax (matrix notation) and find the eigenvalues and eigenvectors. system: x' = -x - 4y - 4 y' = x - y - 6 Can you help? Thanks
  21. T

    Solving for complex eigenvalues

    Quick question: I have a characteristic polynomial: λ2 + i = 0...how do I solve for the eigenvalues? They're suppose to be + or - (√2/2)(1 - i) How'd they get those?
  22. Y

    MHB Which statement about eigenvalues and eigenvectors is not true?

    One more question please... which one of these statements is NOT true (only one can be false): a. a square matrix nXn with n different eigenvalues can become diagonal. b. A matrix that can be diagonal is irreversible. c. Eigenvectors that correspond to different eigenvalues are linearly...
  23. T

    Eigenvalues, eigenvectors, and eigenspaces

    Homework Statement The Attempt at a Solution T(1,0,0) = (3,-1,0) T(0,1,0) = (0,1,0) T(0,0,1) = (-1,2,4) Thus, we have the matrix, \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right| Δ_T (t) = det( \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right|...
  24. matqkks

    Exploring the Power of Eigenvalues and Eigenvectors in Matrix Analysis

    We are aware that by knowing the eigenvalues and eigenvectors we can evaluate the determinant, say if it is invertible and diagonalize to find powers of matrices. Is there a list of properites of a matrix we can find by eigenvalues and eigenvectors? Are there things that e.values and e.vectors...
  25. T

    Eigenvalues of Inverse Transformations

    Homework Statement The Attempt at a Solution So I observed: T(B) = λB T-1(B) = λ'B Also, T-1(T(B)) = λ'λB = B This implies, λ'λ = 1 And so, there should be a relation λ = \frac{1}{λ'}. Is that right?
  26. P

    MATLAB MATLAB: anti-crossing eigenvalues

    I am solving an eigenvalue problem -- Hamiltonian problem in Quantum Mechanics. The matrix is 8x8 with off-diagonal terms, but some are zero. It is well known that the eigenvalues of a Hermitian matrix anti-cross as it nears each other. This is very easy see if the matrix have an independent...
  27. F

    Cauchy stress principle & eigenvalues of stress tensor

    First of all, thanks for all the helpful comments to my previous posts. I'm trying to get a grasp of stress tensors and have been doing some studying. In the literature I've been looking at, it says something about the eigenvalues of stress tensors and the principle stresses. This is...
  28. V

    Finding Charge Conjugation Eigenvalues

    I've just recently been introduced to charge conjugation while reading the introductory particle physics texts by Griffiths and Perkins, and neither one really seem to explain how you go about finding the values for C. For example, if I wanted to find the value for the \rho^0 meson (which I...
  29. T

    Composition of endomorphisms have same eigenvalues

    Homework Statement For two endomorphisms ψ and φ on a vector space V over a field K, show that ψφ and φψ have the same eigenvalues. "Hint: consider the cases λ=0 and λ≠0 separately." The Attempt at a Solution I know that similar endomorphisms (φ and ψφ(ψ^-1)) have the same...
  30. J

    Eigenvalues for Equilibrium Points of First Order Nonlinear DE

    Homework Statement dy/dx = y^3-3y^2+2y it's asking for equilibrium points and for the eigenvalues and stability at each point. Homework Equations The Attempt at a Solution I found the equilibrium points by setting dy/dx = 0 as we were taught to do in class and got y = 0, 1, 2...
  31. S

    Symmetric, irreducible, tridiagonal matrix: Eigenvalues

    Homework Statement A) Let A be a symmetric, irreducible, tridiagonal matrix. Show that A cannot have a multiple eigenvalue. B) Let A be an upper Hessenberg matrix with all its subdiagonal elements non-zero. Assume A has a multiple eigenvalue. Show that there can only be one eigenvector...
  32. Roodles01

    Finding eigenlines & eigenvalues

    In my example I have matrix A = (1 2) . . . . . . . . . . . . . . . . . . . . . . (3 2) Finding the eigenvalue through the method I understand & can get the result i.e. k = 4 & -1 I suspect my algebra is the shaky link, here, but to find the eigenline I find a bit more of a...
  33. J

    Finding the eigenvalues of a complex matrix

    Hi, I am aware of the implicit QR algorithm, which utilises the 'Francis QR step' to find the eigenvalues of a real, square matrix. But, how would one find the eigenvalues of a complex matrix? Would the 'explicit' version of the QR algorithm be used here, using complex arithmetic? Thanks
  34. R

    Wrong values for eigenvalues and frequencies

    Homework Statement I'm working on a problem that involves solving for the eigenvalues and the natural frequencies. I've attached my work as a pdf file and also the MATLAB code used to get the result. The problem that I'm running into is that the frequencies computed from the determinant are...
  35. H

    2 dimensional harmonic oscillator.find the energy eigenvalues?

    Homework Statement Potential of a simple harmonic oscillator is \frac{1}{2}m\omega ^{2}(x^{2}+4y^{2}).Find the energy eigenvalues? Homework Equations Seperation of variables,i think. But i got stuck in the midway. The Attempt at a Solution \frac{-\hslash ^{2}}{2m}\left(...
  36. Shackleford

    Eigenvalues of A-adjoint and A

    Eigenvalues of A* and A Show that the eigenvalues of A* are conjugates of the eigenvalues of A. I know this is an easy problem, but I've just been spinning my wheels manipulating the equations with the transpose, conjugate, and adjoint properties. \begin{align} A^* = \bar{A}^T\\...
  37. Y

    MHB Eigenvalues and Eigenvectors of a Non-Diagonalizable Matrix

    Hello, sorry that I am asking too many questions, I am preparing for an exam... I have a matrix, 0 1 0 0 0 0 0 0 1 and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P) I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the...
  38. D

    Eigenvalues and eigenvectors of J.n

    Homework Statement Calculate the eigenvalues and eigenvectors of the operator, J.n, where n is a unit vector characterized by the polar angles theta and phi, and J is the spin-1 angular momentum operator. Homework Equations Matrix representations for J^2 and J(z) The Attempt at a...
  39. Z

    Angular momentum operator eigenvalues in HO potential.

    Homework Statement Find wave functions of the states of a particle in a harmonic oscillator potential that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually...
  40. K

    Quantum Operators - Eigenvalues & states

    Homework Statement an operator for a system is given by \hat{H}_0 = \frac{\hbar \omega}{2}\left[\left|1\right\rangle\left\langle1\right| - \left|0\right\rangle\left\langle0\right|\right] find the eigenvalues and eigenstates Homework Equations The Attempt at a Solution so i...
  41. L

    Eigenvalues of a rotation matrix

    Homework Statement Find the eigenvalues and normalized eigenvectors of the rotation matrix cosθ -sinθ sinθ cosθ Homework Equations The Attempt at a Solution c is short for cosθ, s is short for sinθ I tried to solve the characteristic polynomial (c-λ)(c-λ)+s^2=0, and...
  42. P

    How can I accurately find eigenvalues for a Jordan canonical form matrix?

    Homework Statement Ok I was working with finding Jordan canonical form... Here is the matrix I was working on: | 1 1 1 | |-1 -1 -1 | | 1 1 0 | I am having problem with finding eigenvalues... below is the attempt to solution I was not getting the right answer. So, when I used...
  43. F

    Eigenvalues and eigenvectors of observables

    Homework Statement Calculate the Eigenvalues and eigenvectors of H= 1/2 h Ω ( ]0><1[ + ]1><0[ ) Homework Equations I know H]λ> = λ]λ> The Attempt at a Solution I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?
  44. C

    What does eigenvalues and eigenvectors mean?

    I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how. What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways? I...
  45. K

    Eigenvalues of a completely disconnected graph

    According to theory the eigenvalues of a completely disconnected graph (no two nodes are connected) must be all 0. But the normalized Laplacian of such a graph will be an identity matrix whose eigenvalues will be all 1s. Please correct me!
  46. K

    Eigenvalues for a 400x400 normalized laplacian of a graph

    This is related to spectral graph theory. I am getting the following eigenvalues for a 400x400 matrix which is a normalized laplacian matrix of a graph. The graph is not connected. So why am i getting a> a negative eigenvalue. b> why is not second eigenvalue 0? ... I used colt(java) and octave...
  47. H

    How do I numerically find eigenvectors for given eigenvalues?

    My aim was to numerically calculate eigenvalues and eigenvectors for a square A matrix. I managed to find the eigenvalues by using QR algorithm. Now, I can find all of the eigenvalues for any given square matrix. But, for the next step, how do I find the corresponding eigenvectors? Is there...
  48. S

    Eigenvalues? DO I have the right idea with this problem?

    Homework Statement I'm supposed to find all the eigenvalues for this 2x2 matrix: 0 2 3 0 Homework Equations The Attempt at a Solution When I tried to do it with no row interchanges, I got the characteristic characteristic equation: λ2-6=0 So, instead of solving this, I...
  49. G

    Eigenvalues of commuting observables (angular momentum)

    Homework Statement Is z|lm\rangle an eigenstate of L^{2} ? If so, find the eigenvalue.Homework Equations L_{z}|lm\rangle = \hbar m|lm\rangle L^{2}|lm\rangle = \hbar^{2} l(l+1)|lm\rangleThe Attempt at a Solution So since L_{z} and L^{2} are commuting observables, they have are...
  50. D

    How to find eigenvalues and eigenvectors for 5x5 matrix

    I got a 5x5 matrix, if use characteristic equation to find the eigenvalues and eigenvectors are very tedious and trouble, so got any method which are easy to calculate?
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