What is Eigen vectors: Definition and 30 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. George Keeling

    How can a diagonalising matrix be unitary?

    This refers to problems A.28/29 from Quantum Mechanics – by Griffiths & Schroeter. I’ve now almost finished the Appendix of this book and been greatly helped with the problems by Wolfram Alpha. In problems A.28/29 we are asked to "Construct the unitary matrix S that diagonalizes T" where T is...
  2. J

    Linearly Independent Eigen Vectors

    Solved (sorry i tried again and realized my E-values were wrong) 1. Homework Statement Homework Equations Find Eigen Values and then what? The Attempt at a Solution I got eigen values as 3 and -3. Now how to proceed? I got Eigen Vector as: 1, 1 for eigen value of 3 and eigen vector as 8, 2...
  3. 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...
  4. S

    I Eigenvectors and inner product

    Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix? I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete. Can two eigenvectors which...
  5. dumbdumNotSmart

    B System of differential equations Basic question

    So I ran into an case I have not seen before. Say we have a system of 3 equations such that W´=AW, where W=(x(t),y(t),z(t)) and A is a 3x3 matrix. The way I usually approach these is by finding the eigenvalues of A to then find the eigenvectors and thus find the ¨homogenous¨ solution. What...
  6. B

    I Eigen Vectors, Geometric Multiplicities and more....

    My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
  7. 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...
  8. 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...
  9. 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?
  10. 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?
  11. B

    Quantum Mechanics, time independant solution in Dirac notati

    Homework Statement Consider the Hamiltonian: $$\hat{H}=C*(\vec{B} \cdot \vec{S})$$ where $C$ is a constant and the magnetic field is given by $$\vec{B} = (0,B,0) $$ and the spin is $$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$ with$$\hat{S}_{x}...
  12. dexterdev

    Physical significance of eigen vectors of Covariance matrix

    Hi all, I have a doubt regarding the physical significance of eigen vectors of the covariance matrix. I came to know that eigen vectors of covariance matrix are the principal components for dimensionality reduction etc, but how to prove it?
  13. Ravi Mohan

    Condition for completeness of eigen vectors of an operator

    I am studying an article http://arxiv.org/abs/quant-ph/9907069 and having some problems understanding it. Is self adjointness of an operator a sufficient or necessary and sufficient requirement for its eigen vectors with the generalized eigenvectors (i don't know what are these) to form...
  14. M

    Find the symmetric matrix from eigen vectors

    Hello to all of you, Is there a way to get the matrix A=[a b c d] from the eigenvectors (orthogonal) matrix H= sin(x) cos(x) cos(x) -sin(x) or to pose it differently to find a matrix that has these 2 eigenvectors ? Thank you in advance . Michael
  15. N

    Undamped 2 DOF vibration. What should the eigen vectors be here

    Normal modes of vibration, two masses, two spring, arranged vertically with m2 at the top, m1 underneath arranged (top to bottom) m2, k2, m1, k1, rigid support I have solved the first part of an undamped coupled spring problem to give m_1m_2 \omega ^ 4 + ((m_1+m_2)k_2+m_2k_1)\omega ^2...
  16. E

    Rank of Matrices and Eigen Vectors

    Homework Statement Find the rank off matrices? i)A=[2 0 9 2; 1 4 6 0; 3 5 7 1 ] 3X4 ii)A=[3 1 4; 0 5 8; -3 4 4; 1 2 4;] 4X3 Find Eigen Vectors and Values of A; A = [3 2 4; 2 0 2; 4 2 3 ] Homework Equations -when det(A) is not equal to zero it will the rank of matrices...
  17. J

    Linearly independent eigen vectors

    Hello everyone, this nxn matrix arises in my numerical scheme for solving a diffusion PDE. M = \left(\begin{array}{cccccccccc}1-\frac{Dk}{Vh} & \frac{Dk}{Vh} & 0 & 0 & & & \ldots & & & 0 \\[6pt] \frac{Dk}{h^2} & 1-2\frac{Dk}{h^2} & \frac{Dk}{h^2} & 0 & & & & & & \\[6pt]0 &...
  18. 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...
  19. J

    Exploiting a the inverse to determine eigen vectors?

    Hello, this is related to my research on RCW(FMM) analysis of light shone onto crossed gratings. main question: can you exploit having matrix A and its inverse A^{-1} to calculate the eigen vectors/values of A in a more effective way? backgroud: so my problem is this: I have an unusual...
  20. D

    Aligning a matrix with its eigen vectors and other questions?

    Hi, I have a square symmetrical matrix A (ugly I know) 321.1115, -57.5311, -33.9206 -57.5311, 296.7836, 10.8958 -33.9206, 10.8958, 382.1050 which has the eigen values, 248.8034 341.6551 409.5415 Am I right in saying that A...
  21. P

    How Do You Compute the Matrix Representation and Check for Eigenbasis?

    For each of the following linear operators T on a vector space V and ordered bases beta, compute [T]beta, and determine whether beta is a basis conisting of eigen vectors of T. V=R^2, T((a,b)^t)= (10a-6b 17a-10b) and beta ={(1,2)^t , (2,3)^t) im...
  22. A

    Are probabilities for Eigen Vectors of Hamiltonians unaffected as time evolves?

    Let a state vector \psi is eigen vector for a Hamiltonian H which governs the Schrodinger equation (in its general form)of a system. Then, will probability distribution of \psi w.r.t any observable remain unchanged as time evolves?
  23. 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 ...
  24. 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...
  25. N

    How can I ensure that Q'*M*Q=I without using MATLAB?

    Homework Statement det(K-lambdaM)=0 K is stiffness and M is mass matrix Homework Equations How do I make sure that Q'*M*Q=I Where Q is the matrix whose column entries are the eigen vectors. The Attempt at a Solution I want to know how to do it without MATLAB help. Is it always...
  26. S

    Eigenvalues and Eigen-vectors for Botched Homework | Characteristic Equation

    Homework Statement (a) Find the characteristic equation and (b) the eigenvalues and corresponding eigen-vectors [-5 0 0 3 7 0 4 -2 3] Homework Equations det(\lambda*I - A) The Attempt at a Solution Finding the characteristic equation wasn't that challenging, I took...
  27. Amith2006

    Eigen functions & eigen vectors

    In Quantum mechanics, we frequently deal with eigen value equations. When we speak of eigen value equations, we come across terms like eigen values,eigen vectors,eigen functions etc. When an operator is operated on certain quantities we get the same quantity multiplied by a constant. These...
  28. B

    Diagonalization & Eigen vectors proofs

    Homework Statement Question 1: A) Show that if A is diagonalizable then A^{T} is also diagonalizable. The Attempt at a Solution We know that A is diagonalizable if it's similar to a diagonal matrix. So A=PDP^{-1} A^{T}=(PDP^{-1})^{T} which gives A^{T}=(P^{-1})^{T}DP^{T} as...
  29. D

    Mixing problem and Eigen vectors relationship.

    Homework Statement A mixing protocol for 5 containers of solvent consists of a repetition of the following procedure. The contents of each container are removed and divided up into pre-defined fractions. The various fractions are then poured back into the containers in a pre-assigned way...
  30. K

    Understanding Eigen Vectors and Diagonalization for a 2x2 Matrix

    I am trying to find eigen values and eigen vectors for A Its 2X2 matrix. A first row (16 -10) second row (-10 24) I got Eigen values as 30.77 and 9.22 but when i try to find eigen vectors here are the equations I end up with -14.77v1 - 10v2= 0 -10v1 - 6.77v2 = 0 Kinda confused how to...