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. 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...
  2. 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?
  3. 3

    What Are the Eigenvalues of A Transpose A?

    Homework Statement Let A be an m x n matrix with rank(A) = m < n. As far as the eigenvalues of A^{T}A is concerned we can say that... Homework Equations The Attempt at a Solution If eigenvalues exist, then A^{T}Ax = λx where x ≠ 0. The only thing I think I can show is that...
  4. B

    Relationship between eigenvalues of 2x2 matrices within a 4x4 matrix

    Homework Statement Consider a 4 x 4 matrix A = B C 0 D where B, C, and D are 2 x 2 matrices. What is the relationship between the eigenvalues of A, B, C, and D? The Attempt at a Solution I suppose you can write A as: b1 b2 c1 c2 b3 b4 c3 c4 0 0 d1 d2 0 0 d3...
  5. G

    (Linear Algebra) Distinct Eigenvalues of a Matrix

    I am reading through a proof and one line of it is not immediately obvious to me, despite it's simplicity. It relates to eigenvalues of a (nearly) full rank, symmetric matrix. Say we have a symmetric matrix A(nxn) that has rank=n-1. Why is this enough to say that all eigenvalues of A are...
  6. B

    Quantum Mechanics, Schrodinger equations and energy eigenvalues

    How do you find an expression for the energy eigenvalues from the TISE (Time Indipendant Schrodinger Equation) for a given potential. e.g. why is: E = (N + 1) hbar*omega an expression for the energy eigenvalues for a potential of: V = 1/2*m*omega2x2 ?? I really have no idea where to start...
  7. S

    Rabbit wolf populations and eigenvalues

    Homework Statement We are initially given the system: \frac{dr}{dt} = 5r -2w \frac{dw}{dt} = r + 2w Initially there are 100 rabbits and 50 wolves. Where the above corresponds to rabbit and wolf populations over time. I solved that system of equations to find the population of rabbits and...
  8. W

    Multiple eigenvalues - any hints would be appreciated

    Homework Statement I need to prove that a 4x4 matrix has 2 zero eignenvalues. 2. The attempt at a solution I have tried to obtain the characteristic equation but calculating the determinant of a relevant 4x4 is rather daunting as there aren't many zeros. I was wondering if there is...
  9. S

    Differential equations with matrices and eigenvalues?

    Homework Statement this is the homework that i have to do http://img690.imageshack.us/img690/2783/problemsb.png Uploaded with ImageShack.us The Attempt at a Solution im not really sure if this is the right method but i will solve it like if it was a homogeneous equation by...
  10. J

    Using eigenvalues and eigenvectors to solve system of ODEs

    Homework Statement Use eigenvalues and eigenvectors to find the general solution of the system of ODEs.. x1 = 3x1 - x2 x2 = -x1 + 2x2 - x3 x3 = -x2 + 3x3 Homework Equations The Attempt at a Solution I converted that into the matrix...
  11. S

    How do eigenvalues and eigenvectors relate to matrices?

    I have to be able to figure out eigenvalues and eigenvectors for 2x2 and 3x3 matrices for my physics course, but I have never taken linear algebra so I obviously have no idea what they even are. I need someone to basically teach me how to solve these problems because I have no knowledge of this...
  12. S

    Evaluating B4: Finding the Eigenvalues & Eigenvectors

    Homework Statement Let B be a matrix with characteristic polynomial λ2-λ√6+3. Evaluate B4. Homework Equations Bn=PDnP-1 The Attempt at a Solution I can find the eigenvalues from the characteristic equation and those would form the diagonal entries of D. But how would I find P, which contains...
  13. U

    Eigenvalues / eigenvectors concept explaination please

    Hello This is a concept question I do not understand. I'm just wondering why the answer is what it is. (the answer is written below the question, I just have no idea where it comes from)
  14. A

    Fortran Fortran90: Subroutine DSYEV and associating eigenvalues and eigenvectors.

    Greetings. I am using the LAPACK (Linear Algebra Package) software package to find the eigenvalues and eigenvectors of a large symmetrical real matrix. Specifically, I calculate a scalar from each eigenvector, and I want to graph it against its associated eigenvalue. I am using the subroutine...
  15. N

    2 Q's on Quantum Stat Mech: why only eigenvalues, why dE

    Hello, Given equilibrium, why does one only consider a mixed state where the pure states are eigenfunctions of the hamiltonian, i.e. states with an energy eigenvalue? And for the second question, I quote David Tong's ``Lectures on Statistical Physics'' ( freely and legally accessible on...
  16. T

    Eigenvalues and Eigenstates of Spin Operator

    I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it. Homework Statement Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,\phi) using the Pauli...
  17. J

    Exponential of jordan form involving distinct eigenvalues

    For a Jordan form (which is a direct sum of individual jordan blocks) that has distinct eigenvalues along the diagonal, how would the exponential of the Jordan form be calculated ? As far as I am aware the formula for calculating the exponential of a jordan block can only involve one...
  18. T

    Why Do Eigenvalues of A+B Equal the Sum of Eigenvalues of A and B?

    Homework Statement I solved a problem that asked me to show that the sum of the eigenvalues of A+B equals the sum of all the individual eigenvalues of A and B, and similarly for products. I just would like to know why is this so... Homework Equations The Attempt at a Solution Is...
  19. L

    Linear Algebra Matrix Eigenvalues

    Homework Statement The matrix A has 3 distinct eigenvalues t1< t2< t3. Let vi be the unique eigenvector associated to ti with a 1 as its first nonzero component. Let D= [t1 0 0 0 t2 0 0 0 t3] and P= [v1|v2|v3] so that the ith column of P is the eigenvector vi...
  20. B

    MATLAB Ordering the eigenvalues in Matlab

    Normally when I work in matlab, the eigenvalues of a matrix are arranged in order from least to greatest when I call the function eig(Matrix). But for some reason, Matlab has decided to start arranging them in order of magnitude (greatest to least)... so that it would arrange the following...
  21. U

    Help Finding Eigenvalues | Quick Answers

    helppp please
  22. T

    Eigenvalues and ground state eigenfunction of a weird Hamiltonian

    Hello again everyone! I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator...
  23. R

    Discrete eigenvalues and their eigenfunctions

    What's the proof that eigenfunctions of discrete eigenvalues are in Hilbert Space?
  24. J

    Same eigenvalues = same jordan form ?

    Say that for two mxm matrices, they have equivalent eigenvalues If this is the case, is it safe to assume that the Jordan forms of both matrices will be the same ? My reasoning comes from the fact that a general jordan block is represented by the following matrix λ 1 0 0 0 λ 1 0 0 0...
  25. J

    Eigenvalues and eigenvectors, 3x3 matrix using Remainder and Factor Theorem

    Homework Statement i = the 3x3 matrix below 2-λ 0 1 -1 4-λ -1 -1 2 0-λ Using remainder and factor theorem find the 3 values of λ. Homework Equations |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3| |a|=ad-bc The Attempt at a Solution(2-λ) |(4-λ x...
  26. T

    Solving Eigenvalues and Eigenfunctions of Hamiltonian

    Hi everyone! I am answering this problem which is about the eigenvalues and eigenfunctions of the Hamiltonian given as: H = 5/3(a+a) + 2/3(a^2 + a+^2), where a and a+ are the ladder operators. It was given that a = (x + ip)/√2 and a+ = (x - ip)/√2. Furthermore, x and p satisfies the...
  27. L

    Eigenvalues and eigenvectors for linear transformation

    Homework Statement V is a vector space consisting all functions f:R->R that is differentiable many times (a) Let T:V->V be the transformation T(f)=f' Find the (real) eigenvectors and eigenvalues of T (b) Let T be transformation T(f)=f" Prove that all real number, m is the eigenvalue of...
  28. DryRun

    Find the eigenvectors given the eigenvalues

    Homework Statement This is the matrix A, which i need to find the eigenvalues and eigenvectors. 3x3 matrix 5 6 12 0 2 0 -1 -2 -2 The attempt at a solution I have found the eigenvalues to be: 1, 2, 2. So, the final eigenvalues are : 1 and 2. Now, i found the eigenvector for...
  29. T

    Why were eigenvalues and eigenvectors defined?

    I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?
  30. C

    Eigenvalues and eigenfunctions

    Homework Statement How does one find all the permissible values of b for -{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y with boundary conditions y(0)=y(1)=0? Thanks. Homework Equations See aboveThe Attempt at a Solution I assume we have a discrete set of \{b_n\} where they can be regarded as...
  31. S

    Compute the Eigenvalues and Eigenvectors

    Homework Statement Compute the Eigenvalues and Eigenvectors of A A= [0 0 1;0 2 0;3 0 0] Homework Equations |A-lamda*I|=0 where I know the lamdas and plug them into the above equation and expand the system of equations. The Attempt at a Solution I have solved for the...
  32. U

    Diagonalization of square matrix if not all eigenvalues are distinct of

    Is it possible to diagonalize such matrix? and how would one do it?
  33. S

    Eigenvalues <1 imply 0 as a limit

    The following question was posed on an old qualifying exam for linear algebra: Suppose A is an n by n complex matrix, and that A has spectral radius <1 (the eigenvalue with largest norm has norm <1). Show that A^n approaches 0 as n goes to infinity. The solution is easy when the eigenspace...
  34. B

    Eigenvalues and eigenvectors of a matrix

    Hello i have this matrix \in Z mod 7, M = \begin{pmatrix} 0&6\\ 5&0 \end{pmatrix} always modulo 7 in Z. I found characteristic polynomial x^2+5. Eigenvalues are \lambda = 3, \lambda' = 4 Eigenvectors related to \lambda = 3 are the non-zero solution of the system: 4x +6y = 0, 5x+4y = 0 I...
  35. H

    Proving Integer Eigenvalues of Matrix A

    Homework Statement Prove: If a, b, c, and d are integers such that a+b=c+d, then A=[a b] [c d] has integer eigenvalues, namely,λ_1{}=a+b and λ_2{}=a-c Homework Equations No relevant equation. The Attempt at a Solution No idea :(
  36. E

    Find the eigenvalues of Liouvillian

    The master equation of the damped harmonic oscillator is \frac{d}{dt}\rho_S(t) = -i\omega_0 [a^\dagger a,\rho_S(t)] + \gamma_0(\bar n+1) \{ a\rho_S(t) a^\dagger -\frac{1}{2} a^\dagger a \rho_S(t) -\frac{1}{2} \rho_S(t) a^\dagger a \} + \gamma_0\bar n \{ a^\dagger \rho_S(t) a...
  37. R

    Solving a System of Coupled DEs: Eigenvalues & Trajectories

    Homework Statement Express y'' + 5y' - 24y = 0 as a system of couple first order DEs, find the eigenvalues of the system and the nature of the critical point at the origin. As well as find the general solution to the system of coupled equations and sketch some trajectories in the phase...
  38. S

    Linear Algebra) If A is similar to inverse of A, must all the eigenvalues equal 1?

    Homework Statement If A is similar to A^(-1) (=inverse of A), must all the eigenvalues equal 1 or -1? Homework Equations The Attempt at a Solution I don't know why the textbook gives me the specific value 1 or -1. If A is similar to its inverse, are the eigenvalues really 1 or...
  39. R

    Finding eigenvalues and eigenvectors of a matrix A

    Homework Statement Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4] Homework Equations det(A - lambda I) = 0 The Attempt at a Solution After expanding using the formula I have the equation (2 - \lambda)(4 - \lambda) - 8 Which gives \lambda = 0, 6 (Should I...
  40. P

    Eigenvalues of a symmetric operator

    I'm reading from Wikipedia: I thought linear operators always had eigenvalues, since you could always form a characteristic equation for the corresponding matrix and solve it? Is that not the case? Are there linear operators that don't have eigenvalues?
  41. M

    What are the eigenvalues of the 3x3 matrix [2 2 1; 1 3 1; 2 2 2]?

    Ive been trying for 3 hours now and can't seem to find the eigenvalues, the long polynomials are getting me confused, the matrix is [2 2 1:1 3 1:1 2 2] So far i did [2-L 2 1:1 3-L 1:1 2 2-L] then I do the normal way to find the determinant but after that I get a horrible polynomial...
  42. H

    Diagonalization of Eigenvalues: A Mistake in Homework Answer?

    Homework Statement I think my teacher made a mistake in his homework answer. I need to verify this for practice. The answer I got is below. The answer the teacher has is in the pdf. Homework Equations Please refer to attached pdf The Attempt at a Solution So there is two...
  43. R

    Effect on eigenvalues of multiplying by a diagonal matrix

    Hi, While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix D^{H}A^{H}AD where the matrix A is known and the matrix D is a diagonal matrix whose entries contain the variables that need to be optimized (those...
  44. S

    Help with eigenvalues from matrix

    Homework Statement find the eignevalues (a part of a larger problem) for A= | -4 1 1 | | 1 5 -1 | | 0 1 -3 | Homework Equations The Attempt at a Solution = | -4-x 1 1 | | 1 5-x -1...
  45. M

    Eigenvalues of Hermitian opertors

    I'm looking for a proof of the fact that orthogonal eigenfunctions of a Hermitian operator have distinct eigenvalues. I know the proof the converse: that eigenfunctions belonging to distinct eigenvalues are orthogonal. thanks alot!
  46. L

    Why are observables represented by operators in Hilbert space?

    i have been trying to learn a bit of quantum mechanics,this is some thing that has been bothering me , if the states of a system can be expressed as vectors in the Hilbert space,what is the motivation behind saying that physical observables can be given by operators?even then ,how can we say...
  47. L

    Eigenvalues of a matrix B= f(A) given eigenvalues of A

    Homework Statement Find the eigenvalues/vectors of A. (I can do this bit :P, A is a 3x3 matrix) What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix? Homework Equations The Attempt at a Solution I have (correctly) found that A...
  48. K

    Mathematica How to make the positions of eigenvalues consistently in Mathematica

    Hi there, Iam just wondering that at different values of m and n, the position of eigenvalues are always varies accordingly. I mean, the outputs positions of eigenvalues are not consistent given by mathematica. Please see attached file for reference. My question is that how to fix this...
  49. S

    Find the eigenvalues of cos(x) -sin(x)

    How do i find the eigenvalues of cos(x) -sin(x) sin(x) cos(x) and cos(x) sin(x) sin(x) -cos(x) Thanks
  50. L

    Differentiability of eigenvalues of a positive matrix

    I have a matrix A, which contains only positive real elements. A is a differentiable function of t. Are the eigenvalues of A differentiable by t?
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