What is Eigenvectors: Definition and 458 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. F

    MHB Set of eigenvectors is linearly independent

    I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?
  2. B

    Orthonormality of a complete set of eigenvectors

    hello How to you rigorously express the orthonormality of a complete set of eigenvectors (|q\rangle)_q of the position operator given that these are necessarily generalized eigenvectors (elements of the distribution space of a rigged hilbert space)? The usual unformal condition \langle...
  3. 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...
  4. Seydlitz

    Extracting eigenvectors from a matrix

    Hello, Homework Statement I want to show that a real symmetric matrix will have real eigenvalues and orthogonal eigenvectors. $$ \begin{pmatrix} A & H\\ H & B \end{pmatrix} $$ The Attempt at a Solution For the matrix shown above it's clear that the charateristic equation will be...
  5. H

    Showing there are no eigenvectors of the annhilation operator

    Homework Statement Show there are no eigenvectors of a^{\dagger} assuming the ground state |0> is the lowest energy state of the system. Homework Equations Coherent states of the SHO satisfy: a|z> = z|z> The Attempt at a Solution Based on the hint that was given (assume there...
  6. 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...
  7. D

    What is the process for finding a complex eigenvector?

    Homework Statement Given A = [ (3,-7),(1,-2) ] and λa = \frac{1}{2} + i \frac{\sqrt{3}}{2} find a single eigenvector which spans the eigenspace. Homework Equations The Attempt at a Solution So I row reduced the matrix to get [(2, -5 + i\sqrt{3}),(0,0 ] and from here we can...
  8. D

    Generalized eigenvectors and differential equations

    Let A be an 3x3 matrix such that A\mathbf{v_1}=\mathbf{v_1}+\mathbf{v_2}, A\mathbf{v_2}=\mathbf{v_2}+\mathbf{v_3}, A\mathbf{v_3}=\mathbf{v_3} where \mathbf{v_3} \neq \mathbb{0}. Let B=S^{-1}AS where S is another 3x3 matrix. (i) Find the general solution of \dot{\mathbf{x}}=B\mathbf{x}. (ii)...
  9. M

    Observables and common eigenvectors

    Homework Statement In a given basis, the eigenvectors A and B are represented by the following matrices: A = [ 1 0 0 ] B = [ 2 0 0 ] [ 0 -1 0] [ 0 0 -2i ] [ 0 0 -1] [ 0 2i 0 ] What are A and B's eigenvalues? Determine [A, B]. Obtain a set...
  10. M

    Eigenvalues and eigenvectors

    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}...
  11. Petrus

    MHB When there is a double root for the eigenvalue, how many eigenvectors?

    Hello MHB, I got one question. If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)? Regards, |\pi\rangle
  12. Superposed_Cat

    Why are wavefunctions represented as eigenvectors?

    Hi I was learning about eigenvectors, inner products, Dirac notation etc. But I don't get why wave functions are represented as eigenvectors?
  13. 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}\\...
  14. 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 \(P(f)\)...
  15. P

    Orthogonal eigenvectors and Green functions

    Hi you all. I have to diagonalize a hermitian operator (hamiltonian), that has both discrete and continuous spectrum. If ψ is an eigenvector with eigenvalue in the continuous spectrum, and χ is an eigenvector with eigenvalue in the discrete spectrum, is correct to say that ψ and χ are always...
  16. WannabeNewton

    Killing fields as eigenvectors of Ricci tensor

    Hi guys! I need help on a problem from one of my GR texts. Suppose that ##\xi^a## is a killing vector field and consider its twist ##\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d##. I must show that ##\omega_a = \nabla_a \omega## for some scalar field ##\omega##, which is equivalent to showing...
  17. 1

    Constructing Lyapunov functions using eigenvectors

    Homework Statement How can I produce a Lyapunov function using the eigenvalues and vectors x'=-x+y y'=-x Homework Equations The Attempt at a Solution So I got the matrix using jacobian and I got the matrix -1 1 -1 0 then i found the eigenvalues to be λ_1= (-1+sqrt3...
  18. R

    Uniqueness of eigenvectors and reliability

    Dear All, In general eigenvalue problem solutions we obtain the eigenvalues along with eigenvectors. Eigenvalues are unique for each individual problem but eigenvectors are not, since the case is like that how we can rely that solution based on the eigenvector is correct. Because if solution is...
  19. 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...
  20. 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...
  21. 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?
  22. F

    Query about eigenvectors of a matrix

    Homework Statement Hi so I have the eigenvalue equation S\vec x = λ\vec x where S = \frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} I have correctly calculated the eigenvalues to be λ=±\frac{\hbar}{2} and the corresponding normalised eigenvectors to be: \hat{e}_{1}=...
  23. M

    Eigen values and Eigenvectors for a special case of a symmetric matrix

    Hey guys if i have a vector x=[x1,x2, ... xn] what are the eigenvectors and eigenvalues of X^T*X ? I know that i get a n by n symmetric matrix with it's diagonal entries in the form of Ʃ xii^2 for i=1,2,3,. . . ,n Thank you in advance once again!
  24. 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...
  25. Z

    How can I rotate phonon eigenvectors to make them parallel?

    Hi, can someone help me about rotating phonon eigenvectors? Say I have a primitive cell with 3 atoms, so for each point in reciprocal space, there are 9 eigen frequency and eigenvectors. Each eigenvector is a 9-dimensional complex vector. I calculated eigenvectors from a transverse branch...
  26. F

    Matrix of eigenvectors, relation to rotation matrix

    So I am given B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find S^{-1}BS, which, with S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1...
  27. Fernando Revilla

    MHB Amanda's question at Yahoo Answers (Eigenvalues and eigenvectors)

    Here is the question: Here is a link to the question: How to I find corresponding eigenvectors? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  28. X

    Finding the eigenvectors of a 2nd multiplicity engenvalue

    Homework Statement I'm given this matrice 2 1 0 1 2 0 0 0 3 and I need to find it's eigenvectors Homework Equations The Attempt at a Solution So I get the eigenvalues to be 1,3,3 with 3 being the one with...
  29. X

    Prove two commutative Hermitian matrices have the same eigenvectors

    Hi, Does anyone know how to prove that two commutative Hermitian matrices can always have the same set of eigenvectors? i.e. AB - BA=0 A and B are both Hermitian matrices, how to prove A and B have the same set of eigenvectors? Thanks!
  30. G

    Finding eigenvectors of general matrix given char. eqn.

    Homework Statement This is my first post, so forgive me if anything's out of order. Assumean operator A satisfies the following equation: 1+2A-A^2-2A^3=0 Find the eigenvalues and eigenvectors for A Homework Equations The Attempt at a Solution So the eigenvalues are +1,-1, and...
  31. S

    When to use RREF in finding Eigenvectors

    Hi, I have a quick question, when should one use RREF in finding eigvenvectors? I've read through some books and sometimes they use them and sometimes they do not. I'm sorry for such a potentially stupid question. Thanks, Zubin
  32. A

    Does it matter which eigenvectors

    If you solve the Schrödinger equation time independent and find a number of stationary position states they are eigenstates. So say uou find the eigen state ψ then c*ψ is also an eigenstate, Does it matter which of these I pick as the eigenstate or is it only the eigenvalue that matters?
  33. J

    Learning Eigenvectors & Linear Differential Equations in 24 Hours

    Hi guys, in my college we have this "Independent Study" component for the Intro Linear Algebra class. Basically, I have to solve 4 questions that go beyond what was covered in class. Sadly, being the procrastinator I am, I haven't looked at it until now and the assignment is due Friday. The...
  34. E

    Solving Problem to Get Unnormalized Eigenvectors in C/Fortran

    Hallo, I am trying to solve the following problem. I need to get eigenvectors of a matrix. I know that there are many subroutines for that in linear algebra packages, for instance in Lapack there is DSPEV, but they all give normalized eigenvectors, while I need the "original" unnormalized ones...
  35. A

    Relationship between several operators and their eigenvectors.

    Homework Statement operators: K=LM and [L,M]=1 α is an eigenvector of K with eigenvalue λ. Show that x=Lα and y=Mα are also eigenvectors of K and also find their eigenvalues. Homework Equations K=LM [L,M]=1 Kα=λα The Attempt at a Solution I tried, but its not even worth...
  36. 5

    Use Lagrange multipliers to find the eigenvalues and eigenvectors of a matrix

    Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...
  37. A

    Finding eigenvectors of a matrix that has 2 equal eigenvalues

    Matrix A= 2 1 2 1 2 -2 2 -2 -1 It's known that it has eigenvalues d1=-3, d2=d3=3Because it has 3 eigenvalues, it should have 3 linearly independent eigenvectors, right? I tried to solve it on paper and got only 1 linearly independent vector from d1=-3 and 1 from d2=d3=3. The method I used...
  38. M

    How do I handle degenerate eigenvalues and eigenvectors in quantum mechanics?

    In Quantum, I ran across the eigenvalue problem. They gave me a matrix, and i was asked to find eigenvalues and then eigenvectors. But the eigenvalues, were degenerate and thus i couldn't find the exact normalized eigenvector. What to do in this case? Shoukd i choose arbitrary values? My...
  39. T

    Eigenvectors and eigenvalues - how to find the column vector

    Hi We have a matrix A (picture), the eigenvalues are λ1 = 4 and λ2 = 1 and the eigenvectors are λ1 : t(1,0,1) λ2 : t1(1,0,2) + t2(0,1,0) I have to examine if there's a column vector v that satifies : A*v = 2 v I would say no there doesn't exist such a column vector v because 2 isn't an...
  40. K

    Normalized Eigenvectors

    Homework Statement An observable is represented by the matrix 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 Find the normalized eigenvectors and corresponding eigenvalues. The Attempt at a Solution I...
  41. C

    Eigenvectors, does order matter?

    I got 2 questions about eigenvectors. Let's say you have an eigenvector [1 0 2]^t. 1. Does the order matter? Like can I change the order to [0 1 2]^t or [1 2 0]^t? 2. It can be any scalar multiple of the vector right? Like I could have [2 0 4] or [4 0 8]
  42. L

    Normalized Eigenvectors of a Hermitian operator

    Hi all Homework Statement Given is a Hermitian Operator H H= \begin{pmatrix} a & b \\ b & -a \end{pmatrix} where as a=rcos \phi , b=rsin \phi I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are: \mid +...
  43. S

    Finding a vector A from given eigen values and eigenvectors

    Homework Statement A matrix A has eigenvectors [2,1] [1,-1] and eigenvalues 2 , -3 respectively. Determine Ab for the vector b = [1,1]. Homework Equations The Attempt at a Solution First I put be as a combination of the two eigenvectors ie 2/3[2,1] -1/3[1,-1] = b...
  44. O

    Equivalence of the nullspace and eigenvectors corresponding to zero eigenvalue

    Suppose a square matrix A is given. Is it true that the null space of A corresponds to eigenvectors of A being associated with its zero eigenvalue? I'm a bit confused with the terms 'algebraic and geometric multiplicity' of eigenvalues related to the previous statement? How does this affect the...
  45. R

    Eigenvectors with at least one positive component

    I am wondering if there is a systematic way to fix the phase of complex eigenvectors. For example e^{i \theta}(1,\omega,\omega^2) where e^{i \theta} is an arbitrary phase and \omega and \omega^2 are the cube roots of unity, is an eigenvector of the cyclic matrix \left(\begin{matrix}0&...
  46. E

    Eigenvalues / Eigenvectors - Is this a legit question to ask?

    -2x + 3y + z = 0 3x + 4y -5z = 0 x -2y + z = -4 Find the characteristic equation, eigenvalues / eigenvectors of the system. I'm given to understand the eigenvalue problem is Ax = (lamba)x, but lamba doesn't exist in the system above. How can I solve for the eigenvalues when there are none?
  47. L

    Projection question with eigenvalues and eigenvectors

    Homework Statement Let v be a non-zero (column) vector in Rn. (a) Find an explicit formula for the matrix Pv corresponding to the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v. (b) What are the eigenvalues and eigenvectors of Pv? Compute the...
  48. J

    Making a eigenvector a linear combination of other eigenvectors

    Homework Statement Write the eigenvector of \sigmax with +1 eigenvalue as a linear combination of the eigenvectors of M. Homework Equations \sigmax = (0,1),(1,0) (these are the columns) The Attempt at a Solution ... Don't know what to do. Can someone show me how to do this using...
  49. C

    Need help understanding eigenvalues and eigenvectors

    Homework Statement Find the principal stresses and the orientation for the principal axis of stress for the following cases of plane stress. σx = 4,000 psi σy = 0 psi τxy = 8,000 psi Homework Equations See picture. The Attempt at a Solution...
  50. C

    Eigenvectors of a symmetric matrix.

    Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there...
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