What is Eigen values: Definition and 47 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. codebpr

    A Does the Maximum Lyapunov exponent depend on the eigenvalues?

    I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
  2. K

    Possible Results and Probabilities of a Measurement of Operator Q

    I first Normalise the wavefunction: $$ \Psi_N = A*\Psi, \textrm{ where } A = (\frac{1}{\sum {|a_n^{'}|^{2}}})^{1/2} $$ $$ \Psi_N = \frac{2}{7}\phi_1^Q+\frac{3}{7}\phi_2^Q+\frac{6}{7}\phi_3^Q $$ The Eigenstate Equation is: $$\hat{Q}\phi_n=q_n\phi_n$$ The eigenvalues are the set of possible...
  3. nomadreid

    I Confused about the spectrum of an observable

    This is a very elementary question, from the beginnings of quantum mechanics. For simplicity, I refer to a finite case with pure states. If I understand correctly, the spectrum of an observable is the collection of eigenvalues formed by the inner product of states and hence equal to...
  4. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors fro the matrix: $$ A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$. Homework Equations Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## . The Attempt at a Solution I've found...
  5. W

    How are Faces Encoded for Image Recognition?

    Hi all, I think I have an idea of how the Mathematical aspects of Face recognition work. But I am curious as to what an eigenvector would be in this respect. I am trying to understand it through finding out how pixels are encoded: What map takes a pixel into a collection of...
  6. F

    Linear algebra matrix to compute series

    Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...
  7. LarryS

    I Eigenvectors - eigenvalues mappings in QM

    In non-relativistic QM, say we are given some observable M and some wave function Ψ. For each unique eigenvalue of M there is at least one corresponding eigenvector. Actually, there can be a multiple (subspace) eigenvectors corresponding to the one eigenvalue. But if we are given a set of...
  8. B

    I How do i find the eigenvalues of this tough Hamiltonian?

    I have this Hamiltonian --> (http://imgur.com/a/lpxCz) Where each G is a matrix. I want to find the eigenvalues but I'm getting hung up on the fact that there are 6 indices. Each G matrix lives in a different space so I can't just multiply the G matrices together. If I built this Hamiltonain...
  9. Adgorn

    I Regarding the linear dependence of eigenvectors

    Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue. However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...
  10. D

    Hermitian and unitary show all its eigen values are ±1

    Homework Statement If a matrix is both Hermitian and unitary show all its eigen values are ±1 Have no idea how to solve ,Have an idea what's hermitian and unitary matrix I know eigen values of hermitian matrix are real and for a unitary matrix it on a unit circle . Thanks
  11. F

    I Third Invariant expressed with Cayley-Hamilton Theorem

    The Cayley-Hamilton Theorem can be used to express the third invariant of the characteristic polynomial obtained from the non-trivial solution of the Eigenvector/Eigenvalue problem. I follow the proof (in Chaves – Notes on Continuum Mechanics) down to the following equation, then get stuck at...
  12. G

    Sum of eigenvectors of linear transformation

    Homework Statement Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2. Homework Equations -Eigenvalues and...
  13. Samuel Williams

    Eigenvalue and eigenvectors, bra-ket

    Question Consider the matrix $$ \left[ \matrix { 0&0&-1+i \\ 0&3&0 \\ -1-i&0&0 } \right] $$ (a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues? (b) Show that the eigenvectors |a1>, |a2>, |a3> form an...
  14. D

    Diagonal Scaling of a 2x2 Positive Definite Matrix

    Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by: $$ A = \begin{bmatrix} {A}_{11} & {A}_{12} \\ {A}_{12} & {A}_{22} \end{bmatrix} $$ And a Matrix ## B ## Given by: $$ B = \begin{bmatrix} \frac{1}{\sqrt{{A}_{11}}} & 0 \\ 0 & \frac{1}{\sqrt{{A}_{22}}}...
  15. Diffie Heltrix

    Norm indueced by a matrix with eigenvalues bigger than 1

    Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...
  16. H

    What is the relationship between eigenvalues and eigenvectors in 3x3 matrices?

    What does it mean when it says eigenvalues of Matrix (3x3) A are the square roots of the eigenvalues of Matrix (3x3) B and the eigenvectors are the same for A and B?
  17. Thor90

    Finding eigenvalues with QR method

    Hi, I am trying to solve the problem of finding eigenvalus for a general square symmetric matrix with the QR algorithm. I have understood that this task is much easier if the matrix is in an Hessemberg form, so I have implemented a function that does that with the Housholder method, but I can't...
  18. K

    Physical significance of Eigenvalues and Eigenvector?

    I want to know what exactly Eigen value imply. What is its Physical significance ? Physical significance of eigen vector? Does eigen value concept apply in signal processing or evalvating frequency response off a system?
  19. F

    E^A matrix power series (eigen values, diagonalizable)

    Homework Statement Find an expression for e^A with the powerseries shown in the image linked Homework Equations I know you have to use eigen values and eigen vectors and a diagonal matrix The Attempt at a Solution What I did was just try to actually multiply out the infinite series given. I...
  20. 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!
  21. V

    Why this paradox in calculating eigen values for T*T ?

    Let T be an operator on the vector space V and let λ1, ... , λn be it's eigen values including multiplicity . Lets find the eigen values for the operator T*T then ( where T* refers to the adjoint operator . <u,v> denotes inner product of u and v ) < Tv , Tv > = < λv, λv > =...
  22. 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...
  23. B

    Repeated Eigen Values and their Eigen Vectors

    I have a matrix and can't seem to get my head around finding all the eigen vectors. The matrix is A: (1 0 0 0 1 0 0 1 0 1 0 0 0 0 1 0) I got the eigen values as: λ1 = 1, λ2 = λ3 = λ4 = 0 For λ1: The eigen vector V1 is (0, 1, 1, 1). For λ2 -> λ4: The only eigen vector I...
  24. S

    Complex Eigen Values: Physical Insight

    Please can anyone provide an insight into what do complex eigen values physically indicate? Vishal
  25. S

    Vector iteration/Power method for eigen values

    I have been reading the Vector iteration/power method for computing the Eigen values and eigen vectors and have got some questions. I shall be grateful if someone can help me. Now, in power method we get the dominant eigen value and corresponding eigen vector. This is followed by deflation...
  26. A

    Real Eigen Values: Proving/Disproving Matrix AB

    I need to either prove or disprove by a counterexample the following proposition: " Let A be an m by n row-stochastic matrix in which all entries are positive real numbers and let B be an n by m column-stochastic matrix with the same feature. Then all eigen values of the m by m matrix AB are...
  27. S

    QR factorization- Eigen values

    I'm attempting to write a code for computing the Eigen values of a real symmetric matrix and I'm using the QR algorithm.I'm referring wiki,Numerical Recipees book and other web serach articles. This is a part of the self-study course I'm doing in Linear Algebra to upgrde my skills. My aim...
  28. T

    Regarding Eigen values Of a Matrix

    Given a square matrix A, the condition that characterizes an eigenvalue, λ, is the existence of a nonzero vector x such that A x = λ x; this equation can be rewritten as follows: Ax=\lambda x (A-\lambda )x = 0 (A-\lambda I )x = 0 -------------------- 1 After the above...
  29. N

    Difference between chemical potential, ionization energy, band gap energy, and Fermi level?

    Kohn-Sham Eigen values? Hi everybody... I have read about Density functional theory and Kohn-Sham theorem, I have found in many references that the Kohn-Sham Eigen values have no physically meaning, except the highest Eigen value has been proved by the Sham and Kohn as the Chemical potential...
  30. V

    Eigen Values of Positive Definite Matrix

    Hi, i was wondering is there a way I can find the eigenvalues of a positive definite matrix. ?
  31. N

    Are Eigen Vectors the Same for Eigen Values of Transpose A and A?

    A is a square matrix over F field if k is the eigen value of A prove that k is eigen value of A^t too and has the same eigen vectors ?? eigen vectors are the solution space P(A) is found by solving (A-kI)x=0 dim P(A)=dim n -dim (ro(a)) rho(a)=rho(a^t)...
  32. D

    Calculating Eigenvalues and Eigenvectors

    Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen2.jpg The second term should be De^( - √(5)t), I made a mistake when writing out the question. The Attempt at a Solution I worked it out to be A = 0, B = -1/5, C = 3/10, D = -3/10 answer is 3.09 Is that correct...
  33. S

    Lagrange to find eigen values and vectors?

    Homework Statement im given a matrix A= 1 -2 ///////////////////////-2 4 im told to find the eigen values and the vectors... but the thing is i have never came across this, i learned lagrange multipliers but never used it to find eigen values and vector.. Homework Equations The...
  34. A

    Can Linear Combinations of Non-Degenerate Eigenstates Form New Eigenstates?

    Well, As far as I know the Schrodinger equation for a H atom is linear and real. Suppose it has two solutions, eigenstates, psi1 and psi2 with non degenerate eigenvalues E1 and E2. It is possible to construct two other states which also are solutions to Schrodinger equation as psi3=...
  35. E

    What are the Eigenvalues and Eigenvectors of a 2x2 Matrix?

    Homework Statement A=[1 0] Calculate [2 3] a) Eigenvalues of A b) Eigenvectors of A c) Eigenvalues and eigenvectors of A^3 The Attempt at a Solution I had no idea what I was doing, but I saw someone attempt one somewhere and used the same method Getting x=3 and 1 for part a)...
  36. C

    Testing Eigen Values: How to Check Accuracy Before Finding Eigen Vectors

    Homework Statement This is probably a really simple question , but how do i test my eigen values to see if there right ,---------- (A-tI)x=0 where t is an eigen value , I know how to test if my eigen vectors are correct but how do i test to see if my eigen values are right ...
  37. A

    Finding eigen values of a 2x2 matrix

    Find the eigenvalues of the following 2x2 matrix: (0 1)/(-1 0) Homework Equations By using the forumla (a-λ)(d-λ) -bc I was able to obtain the following: λ^(2) + 1 = 0 λ^(2) = -1 λ = ± √ (-1) Is thos correct? Also what relevance does this have on the fixed points?
  38. S

    Eigen values for a state and spherical harmonics

    Homework Statement The complete wavefunction for a particular state an atom, is Si(r,theta,phi)=Ne^(-Zr/a_0)(Z/a_0)^3/2sqrt(1/4pi). What is the eigenvalue Lz for this state?Homework Equations see above The Attempt at a Solution This is the last one that I'm having trouble with. I have no...
  39. N

    Linear Operators and Eigen Values

    I'm looking for a good website for understanding Quantum Mechanics (i.e. Time Independent Schrodinger Eq'n, Harmonic Oscillators, Rigid Rotors, etc) The operator is linear if the following is satisfied: A[c*f(x)+d*g(x)]=c*A[f(x)]+d*A[fg(x)], where A = an operator of any kind I'm having...
  40. E

    Eigen vectors and Eigen values

    Hi all, I am a complex systems researcher and I need to have complete knowledge about eigen vectors and eigen values. How does change in dimension affect a point's eigen vector and eigen value? What does principal eigen vector and principal eigen value mean for a point of n-dimension...
  41. I

    Eigen values and cubic roots question

    So I found the characteristic equation of a matrix, and I know the roots of the equation are supposed to be the eigenvalues. However, my equation is: \lambda^3-2\lambda^2 I have double checked different row expansions to make sure this answer is correct. So don't worry about how I came to get...
  42. M

    Eigen Values for k^2-8k+-12=0: (k-2) and (k+3)

    what are the eigen values for k^2-8k+-12=0 i got (k-2) (k+3) as eigen values
  43. N

    Relation between rank and number of non-zero eigen values.

    Hi everyone, I've a simple question (the answer may be so trivial that I really ought to be ashamed for asking!) in elementary matrix theory: "Does there exists any relation between the number of non-zero eigen values of a matrix with its rank?" The matrix is taken to be a general (square, of...
  44. Z

    Find Eigen Values: Solve Char. Equation |A-#I| = 0 of Degree 3

    hello everybody, Consider '#' as lamda. How to find roots(eigen values) of characteristic equation |A-#I| = 0. I know how to find it it using numerical methods. But can anyone please show me how to procced for degree 3 equations. thanks and regards.
  45. J

    Calculating Eigen Values of a Sparse Matrix

    I would like to get the eigen values of a sparse matrix of form [ a, b, c, d, e; 1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0] in matlab/octave I use the following to generate the matrix % filename nfactor.m written by Jeff Sadowski % nfactor finds the...
  46. N

    What does it mean to find eigen values and functions of an infinite well?

    What does it mean to find eigenvalues and eigen functions of an infinite well?
  47. I

    Why the energy eigen values for negative energies are always discrete

    Why the energy eigen values for negetive energies are always discrete while that for positive energies are always continuous? Also what is oscillation theorem?
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