What is Eigen vector: Definition and 16 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


{\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. Z

    Problem with calculating eigen vector for 2*2 Matrix

    Homework Statement r1= 2 7 r2=-1 -6 Homework Equations A-lambda*I=0 (A-lambda*I)*x=0 The Attempt at a Solution I have got following eigen values: lambda1 = -5 and lambda2=1 A-lambdaI matrix is: r1 = 7 7 r2 = -1 -1 and x matrix is: r1 =x r2 =y I can't understand why we have to use...
  2. 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...
  3. 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} \\...
  4. 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...
  5. K

    I Measuring Spin in the Stern Gerlach Experiment

    When we are measuring the spin of the electron in the experiment, we choose the spin property as its eigen state for the measurement. The eigen vectors corresponding to these states could be time dependent. Can we still break the problem into solving time independent Schrodinger Equation and...
  6. R

    How do I find the eigenvalue given unknown rows & eigen vect

    Homework Statement Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. A = 4 4 2 * * * * * * x = 2 -1 10 Given that x is an eigenvector of the matrix A, what is the corresponding eigenvalue? Homework EquationsThe Attempt at a Solution 4−λ 4 2 a...
  7. 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...
  8. R

    Eigenvector of Pauli Matrix (z-component of Pauli matrix)

    I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0, I got...
  9. dexterdev

    What is eigen value, eigen vector etc and what is their physical significance?

    What is eigen value, eigen vector etc and what is their physical significance? -Devanand T
  10. S

    Eigen Vector Proofs: Proving Real Symmetric Matrix M is Positive Definite

    Homework Statement Let M be a symmetric matrix. The eigenvalues of M are real and further M can be diagonalized using an orthogonal matrix S; that is M can be written as M = S^-1*D*S where D is a diagonal matrix. (a) Prove that the diagonal elements of D are the eigenvalues of M...
  11. T

    What is a Right Hand Side Vector in Eigenvector Calculations?

    Homework Statement I've done part A, and part D is easy. I'm stuck with part B. I have no idea what a "right hand side vector" is... Homework Equations The Attempt at a Solution Part A: All eigenvectors are valid. Eigenvalues are 1, 1/2 and 1/3.
  12. S

    Quick Eigen Vector Question: Finding Eigen Vectors for Matrix A | Homework Help

    Homework Statement Hi, for the matrix A = 0 1 0 1 0 0 0 0 2 I have calculated the eigen values, and have successfully calculated the eigen vectors for lamda = -1 and 1. However for...
  13. R

    How to Find Eigenvalues and Eigenvectors for 2x2 Matrices?

    Homework Statement im trying to find the eigen vector for these 2 matrices: A=[0,0;0,8] AND A=[-8,0;0,0] Homework Equations The Attempt at a Solution BACICALLY WHAT IM DOING IS "GUESSING" AT What x1, is then I am coming up wth the solution to x2 once I've made my guess for x1. how...
  14. J

    Eigen function, eigen value, eigen vector

    source: http://en.wikipedia.org/wiki/Eigenvalue what are directional losses? what are its consequences?
  15. I

    How do I find the eigenvector for a 2x2 matrix?

    Homework Statement Find the eigenvector for each of the matrices Homework Equations I have a 2X2 matrice. (4, 2) which are on the top and (2,1) which are on the bottom. I understand how to get the value of the eigen, but I am confused about getting the vector. The Attempt at a...
  16. D

    Energy eigenvalue and eigen vector

    I have some question on energy eigenvalue and eigenfunction help please A particle, mass m , exists in 3 dimensions, confined in the region 0< x < 2L, 0 < y < 3L, 0 < z < 3L a) what are the energy eigenvalues and eigenfunctions of the particle? b) if the particel is a...