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.

View More On Wikipedia.org
Recent contents Top users
  1. 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...
  2. 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...
  3. P

    Proving a property of eigenvalues and their eigenvectors.

    Homework Statement I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of A + kI. The Attempt at a Solution ## A\vec{v}=\lambda\vec{v} ## ## (A+kI)\vec{v}=\lambda\vec{v} ## ## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} -...
  4. kq6up

    General Solution for Eigenvalues for a 2x2 Symmetric Matrix

    Homework Statement From Mary Boas' "Mathematical Methods in the Physical Science" 3rd Edition Chapter 3 Sec 11 Problem 33 ( 3.11.33 ). Find the eigenvalues and the eigenvectors of the real symmetric matrix. $$M=\begin{pmatrix} A & H \\ H & B \end{pmatrix}$$ Show the eigenvalues are real and...
  5. P

    A matrix with Repeated eigenvalues and its corresponding eigenvectors.

    Homework Statement I am asked to find the diagonal matrix of eigenvalues, D, and the matrix of corresponding eigenvectors, P, of the following matrix: \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & -2\\ 0 & 0 & -1 \end{pmatrix} Homework Equations The Attempt at a Solution We just started this topic...
  6. J

    What Are the Eigenlines for the Matrix A = (3 -3; 2 -4)?

    [b]1. Find the corresponding Eigenline A= (3 -3 2 -4) Homework Equations A=(a b c d) k2-(a+d)k+(ad-bc)=0 The Attempt at a Solution k2-(3-(-4))k+(3(-4)-(-3)2)=0 k2+k-6=0 (k+2)(k-3) So k=-2 and k=3 Eigenvector for k=3 (3 -3 2 -4)(x y) = 3(x y) (3x -3y 2x -4y)= (3x...
  7. C

    Finding Eigenvectors and Eigenvalues

    Homework Statement The Matrix A is as follows A= [4 -4 0 2 -2 0 -2 5 3] and has 3 distinct eigenvalues λ1<λ2<λ3 Let Vi be the unique eigenvector associated with λi with a 1 as its first nonzero component. Let D = [ λ1 0 0 0 λ2 0 0 0 λ3] and P=...
  8. M

    MHB Find Eigenvalues for $$y''+\lambda y=0$$

    Hey! :o $$y''+\lambda y =0$$ $$y(0)=0$$ $$y'(0)=\frac{y'(1)}{2}$$ I have to show that the eigenvalues are complex and are given by the relation $\cos{\sqrt{\lambda}}=2$ except from one that is real. The characteristic equation is $m^2+\lambda =0 \Rightarrow m= \pm \sqrt{ \lambda}$$*$...
  9. J

    What are Eigenvectors and Eigenvalues in Relation to Matrices?

    Given a vector ##\vec{r} = x \hat{x} + y \hat{y}## is possbile to write it as ##\vec{r} = r \hat{r}## being ##r = \sqrt{x^2+y^2}## and ##\hat{r} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}##. Speaking about matrices now, the the eigenvalues are like the modulus of a vector and the eigenvectors...
  10. A

    Fortran Finding Eigenvalues & Eigenvectors with Fortran99 for Sparse Matrices

    Hi everybody.. How can i use fortran99 to find the eigenvalues & eigenvectors of sparse matrices? Thanx :)
  11. U

    Finding the eigenvalues of a 3x3 matrix

    Homework Statement A = 7 -5 0 -5 7 0 0 0 -6 Can you please show your method aswell. Every time I try I get the wrong answer. FYI Eigen values are 12.2,-6The Attempt at a Solution so far I got: det = 7-λ -5 0 -5 7-λ 0 0 0 -6-λ Im unsure what to do next. I tried doing...
  12. binbagsss

    Quantum Mechanics , bra-ket , angular momentum eigenkets, eigenvalues

    I have a question on the algebra involved in bra-ket notation, eigenvalues of \hat{J}_{z}, \hat{J}^{2} and the ladder operators \hat{J}_{\pm} The question has asked me to neglect terms from O(ε^{4}) I am using the following eigenvalue, eigenfunction results, where ljm\rangle is a...
  13. T

    MHB Eigenvalues of Laplacian are non-negative

    Hi, I need to learn the following proof and I'm having trouble getting my head round it. Any help would be appreciated. Show that if vector x in R^n with components x=(x1,x2,...,xn), then x.Lx=0.5 sum(Aij(xi-xj)^2) where A is the graphs adjacency matrix, L is laplacian. Then use this result to...
  14. I

    All eigenvalues of a Hermitian matrix are real

    We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?
  15. D

    Finding Zeros of System Function using Eigenvalues

    Hi all - working on this problem wanted to see if anyone had any advice - thanks! As shown in section 4.4, the poles of the system H(z) with state matrices \mathbf{A, b, c^t, } d are given by the eigenvalues of \mathbf{A}. Find: Show that, if d\neq0, the zeros of the system are given by the...
  16. W

    Eigenvalues and eigenkets of a two level system

    Homework Statement The Hamiltonian for a two level system is given: H=a(|1><1|-|2><2|+|1><2|+|2><1|) where 'a' is a number with the dimentions of energy. Find the energy eigenvalues and the corresponding eigenkets (as a combination of |1> and |2>). Homework Equations...
  17. U

    Possible measurement, eigenvalues of eigenfunctions and probabilities

    Homework Statement Suppose the angular wavefunction is ##\propto (\sqrt{2} cos(\theta) + sin (\theta) e^{-i\phi} - sin (\theta) e^{i\phi})##, find possible results of measurement of: (a) ##\hat {L^2}## (b)##\hat {L_z}## and their respective probabilities. Homework Equations...
  18. W

    Momentum eigenvalues and eigenfunctions

    Homework Statement For the following wave functions: ψ_{x}=xf(r) ψ_{y}=yf(f) ψ_{z}=zf(f) show, by explicit calculation, that they are eigenfunctions of Lx,Ly,Lz respectively, as well as of L^2, and find their corresponding eigenvalues. Homework Equations I used...
  19. F

    Find eigenfunctions and eigenvalues of an operator

    Homework Statement \hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} Homework Equations Find eigenfunctions and eigenvalues of this operatorThe Attempt at a Solution It leads to the differential eqn - \frac{{{\hbar...
  20. C

    Arrangement of eigenvalues in a Diagonal matrix

    Homework Statement Is it necessary to arrange the eigenvalues in increasing value order? As shown in the image attached, if I arrange my eigenvalues -2, -1, 1 diagonally, my D would be 2^8 , 1, 1 diagonally. However if i arrange it as, say, -1, 1, -2, my D would be different...
  21. U

    Eigenvalues and Eigenvectors of exponential matrix

    Homework Statement Part (a): Find the eigenvalues and eigenvectors of matrix A: \left( \begin{array}{cc} 2 & 0 & -1\\ 0 & 2 & -1\\ -1 & -1 & 3 \\ \end{array} \right) Part(b): Find the eigenvalues and eigenvectors of matrix ##B = e^{3A} + 5I##. Homework Equations The Attempt at a Solution...
  22. S

    Some Queries about Eigenvalues and vectors

    Hi, I am studying quantum mechanics right now and I can't able to understand some questions about Eigenvalues and Eigenvectors. 1. What does the eigenvalue tell us about the quantum mechanical operators i.e. if we operate a momentum operator on ψ what does the Eigen value of that equation...
  23. G

    Eigenvalues of 2 matrices are equal

    Hi all, I have two matrices A=0 0 1 0 0 0 0 1 a b a b c d c d and B=0 0 0 0 0 0 0 0 0 0 a b 0 0 c d I need to prove that two eigenvalues of A and two eigenvalues of B are equal. I tried to take the determinant of A-λI...
  24. B

    F=ℝ: Normal matrix with real eigenvalues but not diagonalizable

    I am going through Friedberg and came up with a rather difficult problem I can't seem to resolve. If ## F = ℝ ## and A is a normal matrix with real eigenvalues, then does it follow that A is diagonalizable? If not, can I find a counterexample? I'm trying to find a counterexample, by...
  25. A

    What is the physical interpretation of eigenvalues in H?

    Hi All, My question is more from applied quantum mechanics. Suppose I have a 2D conductor(or semiconductor). I use eigenstate representation of hamiltonian in transverse direction and real space representation in longitudinal direction (direction of current flow). Now, 1. Hω=Eω , ω being...
  26. Hercuflea

    How to use QR decomposition to find eigenvalues?

    Homework Statement I need to understand how I would go about using QR decomposition of a matrix to find the matrix's eigenvalues. I know how to find the factorization, just stuck on how I would use that factorization to find the eigenvalues. Homework Equations A=QR where Q is an...
  27. T

    Find T-cyclic subspace, minimal polynomials, eigenvalues, eigenvectors

    Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...
  28. T

    How Do Boundary Conditions Affect Complex Eigenvalues in Differential Operators?

    Find the complex eignevalues of the first derivative operator d/dx subject to the single boundary condition X(0) = X(1). So this has to do with PDEs and separation of variables: I get to the point of using the BC and I am left with an expression: 1 = eλ, this is where my issue falls...
  29. P

    Eigenvalues and stability - question.

    Hi, Homework Statement How may I determine whether a system is stable if its input is equal to its output, hence yielding a system(transfer) function equal to 1? Furthermore, could an eigenvalue zero characterize a stable system? I am attaching three examples where I am asked to determine...
  30. B

    Eigenvector of complex Eigenvalues

    Homework Statement ##A=\begin{bmatrix} 16 &{-6}\\39 &{-14} \end{bmatrix}## Homework Equations The Attempt at a Solution I did ##A=\begin{bmatrix} 16-\lambda &{-6}\\39 &{-14-\lambda} \end{bmatrix}## and got that ##\lambda_1=1+3i## and ##\lambda_2=1-3i## The solution...
  31. B

    Linear Algebra and Eigenvalues

    Suppose A is a diagonlizable nxn matrix where 1 and -1 are the only eigenvalues (algebraic multiplicity is not given). Compute A^2. The only thing I could think to do with this question is set A=PD(P^-1) (definition of a diagonalizable matrix) and then A^2=(PD(P^-1))(PD(P^-1))=P(D^2)(P^-1)...
  32. D

    Creating a matrix with desirable eigenvalues

    Hello, I want to generate a (large) matrix with eigenvalues that are all in a small interval. The relationship between the maximum eigenvalue and minimum eigenvalue should be as small as possible, that's the goal. And the eigenvalues must all be positive. Is there any simple way to do...
  33. S

    Finding the eigenvalues of the spin operator

    1. What are the possible eigenvalues of the spin operator \vec{S} for a spin 1/2 particle? Homework Equations I think these are correct: \vec{S} = \frac{\hbar}{2} ( \sigma_x + \sigma_y + \sigma_z ) \sigma_x = \left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right),\quad...
  34. P

    Dif.eq.system; complex eigenvalues

    Homework Statement Given system: dx/dt=-x-5y dy/dt=x+y Homework Equations The Attempt at a Solution So I calculated that \lambda_1=-2i and \lambda_2=2i Generaly \lambda=+-qi next i know that general solution is in form: x=C1cos(qt)+C2sin(qt) y=C*1cos(qt)+C*2sin(qt) So...
  35. Jalo

    Find the eigenvalues of the Hamiltonian - Harmonic Oscillator

    Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...
  36. L

    On SUSY irreps: eigenvalues of Pauli-Lubanski operator?

    Hi everyone, Just an easy question that came to my mind while studying basics of SUSY. Consider in N=1, D=4 a massive clifford vacuum |m,s,s_3\rangle, and for cconcreteness take its spin to be s=1/2. Now, acting with the four supercharges on both the |m,1/2,1/2\rangle and |m,1/2,-1/2\rangle...
  37. M

    How Do Eigenvalues and Eigenvectors Connect to Fourier Transforms?

    Hello guys, is there any way someone can explain to me in resume what eigen values and eigenvectors are because I don't really recall this theme from linear algebra, and I'm not getting intuition on where does Fourier transform comes from. my teacher wrote: A\overline{v} = λ\overline{v}...
  38. H

    Factoring 3rd degree polynomial for eigenvalues

    Homework Statement Was given a matrix To find the eigenvalues I set up the characteristic equation [-1-x | 7 | -5 ] [-4 | 11-x | -6 ] [-4 | 8 | -3-x] With some dirty work I got this bad boy out, which I'm having trouble factoring -x3+7x2-15x+9Homework Equations...
  39. V

    Spring-Mass System: Eigenvalues and Eigenvectors

    The det. of the following matrix: $$ \begin{matrix} 2k-ω^{2}m_{1} & -k\\ -k & k-ω^{2}m_{2}\\ \end{matrix} $$ must be equal to 0 for there to be a non-trivial solution to the equation: $$(k - ω^{2}m)x =0$$ Where m is the mass matrix: $$ \begin{matrix} m_{1} & 0\\ 0& m_{2}\\...
  40. Sudharaka

    MHB Eigenvalues and Eigenvectors over a Polynomial Ring

    Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...
  41. S

    Prove Eigenvalues of an (operator)^2 are real and positive

    Q: Using Dirac notation, show that if A is an observable associated with the operator A then the eigenvalues of A^2 are real and positive. Ans: I know how to prove hermitian operators eigenvalues are real: A ket(n) = an ket(n) bra(n) A ket(n) = an bra(n) ket(n) = an [bra(n) A ket(n)]* =...
  42. Sudharaka

    MHB Eigenvalues of a Linear Transformation

    Hi everyone, :) Here's a question I got stuck. Hope you can shed some light on it. :) Of course if we write the matrix of the linear transformation we get, \[A^{t}.A=\begin{pmatrix}a_1^2 & a_{1}a_2 & \cdots & a_{1}a_{n}\\a_2 a_1 & a_2^2 &\cdots & a_{2}a_{n}\\.&.&\cdots&.\\.&.&\cdots&.\\a_n...
  43. C

    Question about QM eigenvalues.

    I've been wrestling with this question for a while and can't seem to find anything in my notes that will help me. Homework Statement Determine whether the wave function \Psi (x,t)= \textrm{exp}(-i(kx+\omega t)) is an eigenfunction of the operators for total energy and x component of momentum...
  44. F

    Why eigenvalues of L_x^2 and L_z^2 identical?

    Homework Statement Calculate the eigenvalues of the L_x^2 matrix. Calculate the eigenvalues of the L_z^2 matrix. Compare these and comment on the result. Homework Equations L_x=\frac{1}{2}(L_+ + L_- ) The Attempt at a Solution I have derived eigenvalues for each: 0 and \hbar^2...
  45. B

    Getting Eigenvalues Into a Differential Operator

    Following Butkov, a second order ode A(x)y'' + B(x)y' + C(x)y = D(x) can always be brought into Sturm-Liouville form \tfrac{d}{dx}[p(x)y'] - s(x)y = f(x) after multiplying across by H(x) = - \tfrac{1}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}. He then says the function s(x) can...
  46. G

    Eigenvalues of two matrices are equal

    Hi everyone, I have two matrices A and B, A=[0 0 1 0; 0 0 0 1; a b a b; c d c d] and B=[0 0 0 0; 0 0 0 0; 0 0 a b; 0 0 c d]. I have to proves theoretically that two of the eigenvalues of A and B are equal and remaining two eigenvalues of A are 1,1. I tried it by calculating the...
  47. Jameson

    MHB Prove matrix has all real eigenvalues

    Problem: Let $A$ be a $n \times n$ matrix with real entries. Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real. Solution: I'm definitely not seeing how to approach this problem. I know that to calculate the eigenvalues of a matrix I need to solve $\text{det...
  48. J

    Physical significance of eigenvalues?

    Standard Pauli spin matrices are: Sx: $$\left(\begin{array}{cc}0&1/2\\1/2&0\end{array}\right)$$ Sz: $$\left(\begin{array}{cc}1/2&0\\0&-1/2\end{array}\right)$$ The Sz eigenvectors are Z+ = (x=1,y=0) and Z- = (x=0,y=1). These yield eigenvalues 1/2 and -1/2 respectively. Similarly...
  49. S

    Eigenvalues of perturbed matrix. Rouché's theorem.

    Homework Statement Let \lambda_0 \in \mathbb{C} be an eingenvalue of the n \times n matrix A with algebraic multiplicity m , that is, is an m-nth zero of \det{A-\lambda I} . Consider the perturbed matrix A+ \epsilon B , where |\epsilon | \ll 1 and B is any n \times n matrix...
  50. Fernando Revilla

    MHB Eigenvalues of similar matrices

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
Back
Top