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. 1

    Eigenvalues of operator in dirac not* (measurement outcomes)

    Homework Statement A measurement is described by the operator: |0⟩⟨1| + |1⟩⟨0| where, |0⟩ and |1⟩ represent orthonormal states. What are the possible measurement outcomes? Homework Equations [/B] Eigenvalue Equation: A|Ψ> = a|Ψ> The Attempt at a Solution Apologies for the basic...
  2. P

    2n x 2n matrices without real eigenvalues

    Homework Statement For an arbitrary positive integer ##n##, give a ##2n## x ##2n## matrix ##A## without real eigenvalues. Homework EquationsThe Attempt at a Solution First of all, I am having some trouble interpreting this problem. I do not know if it is generalized where I am supposed to find...
  3. M

    MHB Finding Eigenvalues for the Heat Equation: A Step-by-Step Approach

    Hey! :o Find the solution of the problem $$u_t(x, t)-u_{xx}(x, t)=0, 0<x<1, t>0 \tag {*} \\ u(0, t)=0, t>0 \\ u_x(1,t)+u_t(1,t)=0, t>0$$ I have done the following: We are looking for solutions of the form $$u(x, t)=X(x) \cdot T(t)$$ $$u(0, t)=X(0) \cdot T(t)=0 \Rightarrow X(0)=0 \\ X'(1)...
  4. T

    Elementary quantum spin in Sakurai

    Homework Statement I am currently working on a seemingly straightforward eigenvalue problem appearing as problem 1.8 in Sakurai's Modern QM. He asks us to find an eigenket \vert\vec S\cdot\hat n;+\rangle with \vec S\cdot\hat n\vert\vec S\cdot\hat n;+\rangle = \frac\hbar 2\vert\vec S\cdot\hat...
  5. D

    Eigenvalues and diagonalization of a matrix

    When you diagonalize a matrix the diagonal elements are the eigenvalues but how do you know which order to put the eigenvalues in the diagonal elements as different orders give different matrices ? Thanks
  6. rayne1

    MHB Eigenvector of 3x3 matrix with complex eigenvalues

    Matrix A: 0 -6 10 -2 12 -20 -1 6 -10 I got the eigenvalues of: 0, 1+i, and 1-i. I can find the eigenvector of the eigenvalue 0, but for the complex eigenvalues, I keep on getting the reduced row echelon form of: 1 0 0 | 0 0 1 0 | 0 0 0 1 | 0 So, how do I find the nonzero eigenvectors of the...
  7. C

    Numerical solution to Schrödinger equation - eigenvalues

    Not sure whether to post this here or in QM: I trying to numerically solve the Schrödinger equation for the Woods-Saxon Potential and find the energy eigenvalues and eigenfucnctions but I am confused about how exactly the eigenvalues come about. I've solved some differential equations in the...
  8. M

    The Cooper pair box Hamiltonian in the matrix form

    Hello, In my problem I need to We are advised to create the Cooper pair box Hamiltonian in a matrix form in the charge basis for charge states from 0 to 5. Here is the Hamiltonian we are given H=E_C(n-n_g)^2 \left|n\right\rangle\left\langle...
  9. L

    Eigenvalues of positions in atomic orbitals

    Let's take a hydrogen atom with a single electron. How many eigenvalues of position can it form (assuming you put it the atom in an x, y, z coordinate)? like 1 billion possible position eigenvalues? Is it continuous number like 1.1, 1.2, 1.3 or quantized? and either case, how many eigenvalues...
  10. diegzumillo

    Decompose wave packet into eigenvalues of L2, Lz and k

    Homework Statement The free particle wave packet in question is $$\psi=ce^{-(r/r_0)^2}$$ Homework EquationsThe Attempt at a Solution I've been going through books and class notes but I really have no idea where this came from. I'm thinking that if I can decompose this in plane waves I could...
  11. L

    What Are the Eigenvalues of Entangled Electron Spins?

    Lets' say you have an entangled pair of electrons with spin up and spin down. What is its eigenvalues.. is it.. Eigenvalue 1: Electron A with spin up Eigenvalue 2: Electron A with spin down Eigenvalue 3: Electron B with spin up Eigenvalue 4: Electron B with spin down But it's supposed to be...
  12. nomadreid

    Zero eigenvalues or eigenvectors

    I have a bit of problem with zero eigenvectors and zero eigenvalues. On one hand, there seems to be nothing in the definition that forbids them, and they even seem necessary to allow because an eigenvalue can serve as a measurement and zero can be a measurement, and if there is a zero eigenvalue...
  13. M

    Energy Eigenvalues for Ion with Spin

    Homework Statement An ion has effective spin ħ. The spin interacts with a surrounding lattice so that: Hspin = A S2 z. I first had to write H as a matrix. Then i had to find the energy eigenvalues. Homework EquationsThe Attempt at a Solution I figured j=1 and mj = 1,0,-1 S2 z = ħ2(1 0 0; 0...
  14. I_am_learning

    Eigenvalues of Gramian Matrix

    if x is a column vector, then a matrix G = x*xT is a Gramian Matrix. When I tried calculating the matrix G and its eigenvalues for cases when x = [x1 x2]' and [x1 x2 x3]' by actually working out the algebra, it turned out (if I didn't do any mistakes) that the eigen values are all zeros except...
  15. W

    Eigenvalues for a bounded operator

    Homework Statement Let C be the composition operator on the Hilbert space L_{2}(\mathbb{R}) with the usual inner product. Let f\in L_{2}(\mathbb{R}), then C is defined by (Cf)(x) = f(2x-1), \hspace{9pt}x\in\mathbb{R} give a demonstration, which shows that C does not have any eigenvalues...
  16. K

    Number of eigenvalues of this Hermitian

    Hi. I'm trying to study QM from Shankar on my own. Asking this here because I don't really have a teacher to help me with this: Homework Statement I'm trying to solve problem 1.8.9 -part 3 of "The Principles of Quantum Mechanics" by R Shankar. Here's the problem: Given the values of Mij (see...
  17. M

    Eigenvalues of Laplacian Matrix

    hi pf! I am reading a text and am stuck at a part. this is what is being said: If ##g## is a graph we have ##L(g) + L(\bar{g}) = nI - J## where ##J## is the matrix of ones. Let ##f^1,...f^n## be an orthogonal system of eigenvectors to ##L(g) : f^1 = \mathbb{1}## and ##L(g)f^i = \lambda_i...
  18. Y

    MHB Help! Wrong Eigenvalues Using Matrix A

    Hello all, I have a problem with eigenvalues. I tried finding eigenvalues and eigenvectors of a matrix A. I did once using: \[\lambda I-A\] And a second time using: \[A-\lambda I\] For the first eigenvalue I got identical eigenvectors in both methods, but for the second eigenvalue, the...
  19. S

    Question about spin operators and eigenvalues

    I've been watching Leonard Susskind's videos on quantum entanglements. Naturally, one of the things that he has been discussing is spin and its various operator Hermitian matrices and eigenvalues. Now I have two main questions about this: 1. I know that if you apply a spin operator σ (which is...
  20. M

    MHB Inverse Eigenvalues: A Puzzling Question?

    Hey! :o Does it stand that the eigenvalues of $A^{-T}A^{-1}$ are the inverse of the eigenvalues of $A^TA$ ?? (Wondering)
  21. nomadreid

    Continuous set of eigenvalues in matrix representation?

    Let's see if I have this straight: Observables are represented by Hermitian operators, which can be, for some appropriate base, represented in matrix form with the eigenvalues forming the diagonal. Sounds nice until I consider observables with continuous spectra. How do you get something like...
  22. K

    Little question about eigenvalues

    Hey there, I'm thinking about if one of the eigenvalues is zero (means determinant is 0. right?) So, is there any possibility to non-zero eigenvalue also exists?
  23. I

    Finding the unit vector for an ellipse

    Homework Statement Given the ellipse ##0.084x^2 − 0.079xy + 0.107y^2 = 1 ## Find the semi-major and semi-minor axes of this ellipse, and a unit vector in the direction of each axis. I have calculated the semi-major and minor axes, I am just stuck on the final part. Homework Equations this...
  24. T

    Abstract Linear Algebra: Eigenvalues & Eigenvectors

    Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...
  25. D

    Differential Equations System Solutions

    Homework Statement Consider the initial value problem for the system of first-order differential equations y_1' = -2y_2+1, y_1(0)=2 y_2' = -8y_1+2, y_2(0)=-1 If the matrix [ 0 -2 -8 0 ] has eigenvalues and eigenvectors L_1= -4 V_1= [ 1...
  26. T

    Angular Momentum squared operator (L^2) eigenvalues?

    Homework Statement Find the eigenvalues of the angular-momentum-squared operator (L2) for hydrogen 2s and 2px orbitals... Homework Equations Ψ2s = A (2-r/a0)e-r/(2a0) Ψ2px = B (r/a0)e-r/(2a0) The Attempt at a Solution If I am not wrong, is the use of L2 in eigenfunction L2Ψ = ħ2 l(l+1) Ψ...
  27. A

    Comp Sci Eigenvalues and eigenvectors of a real symmetric matrix in Fortran

    Homework Statement I try to run this program, but there are still some errors, please help me to solve this problems Homework EquationsThe Attempt at a Solution Program Main !==================================================================== ! eigenvalues and eigenvectors of a real...
  28. T

    Solving a System of Differential Equations with Complex Eigenvalues

    1. Homework Statement http://puu.sh/cSK1u/62e2f1c74d.png olve the system: x' = [-4, -4 4, -4] with x(0) = [ 2, 3] Find x1 and x2 and give your solution in real form.2. Homework Equations 3. The Attempt at a Solution Just a note here, I'm basically forced to self-learn this course because...
  29. R

    Question about the best method to use for finding wavefunctions and eigenvalues

    We have been covering the annhilation and creation operators in class. You can use the annihilation operator to find the groundstate wavefunction, and then use the hamiltonian in terms of annhiliation and creation operators to find the energy eigen value of that state. (or you could put the...
  30. tasos

    Given HamiltonianFind eigenvalues and eigenfunctions

    Homework Statement We have the hamiltonian H = al^2 +b(l_x +l_y +l_z) where a,b are constants. and we must find the allowed energies and eigenfunctions of the system. Homework EquationsThe Attempt at a Solution [/B] I tried to complete the square on the given hamiltonian and the result is: H =...
  31. V

    Expecting the possible event of zero probability

    Consider a potential well in 1 dimension defined by $$ V(x)= \begin{cases} +\infty &\text{if}& x<0 \text{ and } x>L\\ 0 &\text{if} &0\leq x\leq L \end{cases} $$ The probability to find the particle at any particular point x is zero. $$P(\{x\}) = \int_S \rho(x)\mathrm{d}x=0 ;\forall\; x \in...
  32. S

    [QM] Conservation probability to degenerate eigenvalues

    Hello =) I have a question regarding the conservation of probability in quantum mechanics. We know that the probability of a measurement of a given observable, is preserved in time if the observable commutes with the Hamiltonian. But this is also true if the value of the measurement...
  33. jfy4

    Lower bounds on energy eigenvalues

    Hi, I'm interested in learning about what would be the compliment to the Variational method. I'm aware that the Variational method allows one to calculate upper bounds, but I'm wondering about methods to calculate lower bounds on energy eigenvalues. And for energies besides the ground state if...
  34. F

    Hermitian Operators Eigenvalues

    Homework Statement I have a hermitian Operator A and a quantum state |Psi>=a|1>+b|2> (so we're an in a two-dim. Hilbert space) In generally, {|1>,|2>} is not the eigenbasis of the operator A. I shall now show that the Eigenvaluse of A are the maximal (minimal) expection values <Psi|A|Psi>.The...
  35. F

    Spectrum of the Reduced matrix's eigenvalues

    I would like to know if the density matrix spectrum is always discrete or if it is possible it has a continuum spectrum. It is clear that a pure density matrix has a discrete spectrum but it is not obvious in general. I have heard that all compact operator has discrete eigenvalues and if it has...
  36. tasos

    Finding Energy Eigenvalues and Eigenfunctions for a Particle Well

    Homework Statement (a) Find the energy eigenvalues and eigenfunctions for this well. (b) If the particle at time t = 0 is in state Ψ = constant (0 <x <L)). Normalize this state. Find the state that will be after time t>0 (c) For the previous particle, if we measure the energy at time t = 0...
  37. J

    Function scales eigenvalues, but what happens to eigenvectors?

    Statement: I can prove that if I apply a function to my matrix (lets call it) "A"...whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers...
  38. T

    Eigenvalues with an added part

    Homework Statement Hi, I have an electrical circuit, from which I have derived 4 equations to work out the current I of the circuit. To solve I need to put the equations into a matrix and find the eigenvalues & vectors, great I can do that. However there is an additional matrix on the end for...
  39. A

    Eigenvalues and Eigenvectors of a Hermitian operator

    Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...
  40. R

    Eigenvalues and operators, step the involves switching and substitutin

    Operator C = I+><-I + I-><+I Wavefunction PSI = Q I+> +V I-> C PSI = Q I-> + V I+> note the I is just a straight line (BRAKET vectors), the next step is where I get confused, p is subbed in and the ket vectors switch places... C PSI = pQ I+> + pV I-> <---- why?? therefore V = pQ and Q =...
  41. S

    Eigenvalues of Invertible Matrix

    Homework Statement If A is an invertible nxn matrix, then A has n distinct eigenvalues. (TRUE/ FALSE) Homework Equations The Attempt at a Solution True? We weren't really taught the concept of eigenvalues too well, but from what I can gather square matrices appear to have the...
  42. kini.Amith

    Eigenvalues containing the variable x

    Homework Statement The wave function ψ(x)=Ae-b2x2/2 where A and b are real constants, is a normalized eigenfunction of the schrodinger eqn for a free particle of mass m and energy E. Then find the value of E Homework Equations The Attempt at a Solution Substituting the wave...
  43. P

    Eigenvalues and eigenvectors of a non-symmetric matrix?

    I have a non symmetric matrix AB where A and B are symmetric matrices. How can I find the eigenvectors and eigenvalues of AB? In a paper( Fisher Linear Discriminant Analysis by M Welling), the author asks to find eigenvalues and eigenvectors of B^(1/2)* A *B^(1/2) which is a symmetric...
  44. D

    Momentum and energy eigenvalues

    Hi. I will give you a question I have looked at and then tell you where I am confused. The wavefunction for a particle of mass m is ψ(x) = sin(kx)exp(-iωt) where k is a constant. (i) Is this particle in a state of defined momentum ? If so , determine its momentum. (ii) Is this particle in...
  45. fluidistic

    Can the Operator n dot L Have the Same Eigenvalues as Lz in Quantum Mechanics?

    Homework Statement I've been told that the operator ##\hat n \cdot \hat {\vec L}## has the same eigenvalues as ##L_z##. Later I've been told that it has the same eigenvalues as any component of ##\hat {\vec L}##. But I am a bit confused, as far as I understand the eigenvalues of ##L_x##...
  46. carllacan

    Creation operator and eigenvalues

    Homework Statement Prove that the creation operator a_+ has no eigenvalues, for instance in the \vert n \rangle . Homework Equations Action of a_+ in a harmonic oscillator eigenket \vert n \rangle : a_+\vert n \rangle =\vert n +1\rangle The Attempt at a Solution Calling a the...
  47. M

    MHB The eigenvalues are real and that the eigenfunctions are orthogonal

    Hey! :o We have the Sturm-Liouville problem $\displaystyle{Lu=\lambda u}$. I am looking at the following proof that the eigenvalues are real and that the eigenfunctions are orthogonal and I have some questions... $\displaystyle{Lu_i=\lambda_iu_i}$ $\displaystyle{Lu_j=\lambda_ju_j...
  48. J

    Eigenvalues / Eigenvectors relationship to Matrix Entries Values

    Hi, folks I have had a hard time to find out whether or not there is a theorem in Linear Algebra or Spectral Theory that makes any strong statement about the relationship between the entries of a Matrix and its Eigenvalues and Eigenvectors. Indeed, I would like to know how is the...
  49. kq6up

    What Went Wrong with Imaginary Eigenvalues?

    Homework Statement Multiply the matrices to find the resultant transformation. $$x\prime =2x+5y\\ y'=x+3y $$ and $$ x\prime \prime =x\prime -2y\prime \\ y\prime \prime =3x\prime -5y\prime $$ Homework Equations $$Mr=r\prime$$ The Attempt at a Solution I get imaginary eigenvalues of -i and...
  50. kq6up

    State vectors and Eigenvalues?

    If I define a state ket in the traditional way, Say: $$|\Psi \rangle =\sum _{ i }^{ }{ a_{ i }|\varphi _{ i }\rangle \quad } $$ Where $$a_i$$ is the probability amplitude. How does: $$\hat {H } |\Psi \rangle =E|\Psi \rangle $$ if the states of $$\Psi$$ could possibly represent states...
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