What is Eigenvectors: Definition and 459 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. T

    I Unravelling Structure of a Symmetric Matrix

    Hey guys, I was wondering if anyone had any thoughts on the following symmetric matrix: $$\begin{pmatrix} 0.6 & 0.2 & -0.2 & -0.6 & -1\\ 0.2 & -0.2 & -0.2 & 0.2 & 1\\ -0.2 & -0.2 & 0.2 & 0.2 & -1\\ -0.6 & 0.2 & 0.2 & -0.6 & 1\\ -1 & 1 & -1 & 1 & -1 \end{pmatrix} $$ Notably, when one solves for...
  2. Z

    Does there exist a 2x2 non-singular matrix with only one 1d eigenspace?

    Before going through calculations/reasoning, let me summarize what my questions will be - In order to obtain the desired matrix, I impose five constraints on ##a,b,c,d,## and ##\lambda##. - These five constraints are four equations and an inequality. I am not sure how to work with the...
  3. N

    I Is there always a matrix corresponding to eigenvectors?

    I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?
  4. M

    I Help in understanding Eigenvectors please

    Hi; struggling a little with eigenvectors; I can get to the equation at the foot of the example but I can't understand the "formula" leading to the setting of x = 3 at the foot of the example? thanks martyn
  5. C

    Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

    For exercise 3 (2), , The solution for finding the eigenvector is, However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here? Many Thanks!
  6. C

    Finding eigenvalues and eigenvectors given sub-matrices

    For this, The solution is, However, does someone please know what allows them to express the eigenvector for each of the sub-matrix in terms of t? Many thanks!
  7. M

    I Find the Eigenvalues and eigenvectors of 3x3 matrix

    Assume a table A(3x3) with the following: A [ 1 2 1 ]^T = 6 [ 1 2 1 ]^T A [ 1 -1 1 ]^T = 3 [ 1 -1 1 ]^T A [ 2 -1 0]^T = 3 [ 1 -1 1]^T Find the Eigenvalues and eigenvectors: I have in mind to start with the Av=λv or det(A-λI)v=0.... Also, the first 2 equations seems to have the form Av=λv...
  8. L

    Mathematica Matrices in Mathematica -- How to calculate eigenvalues, eigenvectors, determinants and inverses?

    Hi, In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment. Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...
  9. George Wu

    A What is a spatial wavefunction in QFT?

    My understanding is: $$\phi (\mathbf{k})=\int{d^3}\mathbf{x}\phi (\mathbf{x})e^{-i\mathbf{k}\cdot \mathbf{x}}$$ But what is ##\phi (\mathbf{x})## in Qft? In quantum mechanics, $$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> =\int{d^3}\mathbf{k}\phi...
  10. C

    Diagonalizing a Matrix: Understanding the Process and Power of Matrices

    For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!
  11. C

    Why can't we define an eigenvalue of a matrix as any scalar value?

    For this, Dose anybody please know why we cannot say ##\lambda = 1## and then ##1## would be the eigenvalue of the matrix? Many thanks!
  12. Kekeedme

    I Finding Eigenvectors of 2-state system

    In Cohen-Tannoudji page 423, they try to teach a method that allows to find the eigenvectors of a 2-state system in a less cumbersome way. I understand the steps, up to the part where they go from equation (20) to (21). I understand that (20) it automatically leads to (21). Can someone please...
  13. H

    How to find the eigenvector for a perturbated Hamiltonian?

    Hi, I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian. However, I don't see how to find the eigenvectors. To find the eigenvalues for the first excited state I build this matrix ##...
  14. H

    If |a> is an eigenvector of A, is f(B)|a> an eigenvector of A?

    Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...
  15. H

    I Proof that if T is Hermitian, eigenvectors form an orthonormal basis

    Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...
  16. H

    Prove that ##\lambda## or ##-\lambda## is an eigenvalue for ##T##.

    The statement " If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2##", means that there exists an ##x## in ##V## such that ## T^2 (x) = \lambda^2 x##. If ##T(x) = \mu x##, we've have $$ T [T(x)]= T ( \mu x)$$ $$ T^2 (x) = \mu^2 x$$ $$ \lambda ^2 = \mu ^2...
  17. A

    Question regarding eigenvectors

    So I have been studying for my upcoming math exam and a lot of the problems require to find eigenvalues/eigenvectors.Now the question I have is the following; Take a look at this matrix $$ \left[ \begin{matrix} 6 & -3 \\\ 3 & -4 \end{matrix} \right] $$ Now the eigenvalues are...
  18. U

    I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

    Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
  19. F

    Problem calculating eigenvalues and eigenvectors

    Hello everyone. I am trying to construct a functioning version of randomfields (specifically 2D_karhunen_loeve_identification_example.py) in Matlab. For that, I have to calculate the Karhunen-Loève expansion of 2D data, since this is what it says in the documentation. I also have some sample...
  20. StenEdeback

    Eigenvalues and eigenvectors of J3

    The J3 matrix of two dimensional SU2 consists of two row vectors (1 0) and (0 -1). When I calculate the eigenvalues of an eigenvector v the usual way with J3v=kv then I find eigenvalues +-1 and eigenvectors (1 0) and (0 1). But how is it possible to say that there are other eigenvectors and...
  21. W

    MHB Solving Matrix A: Characteristic Equation and Eigenvectors

    good evening everyone! Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :) Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...
  22. Andrew1235

    Finding the directions of eigenvectors symmetric eigenvalue problem

    In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
  23. David Koufos

    How do I know if my eigenvectors are right?

    For ##M = \begin{pmatrix} 2 & 2\\ 2 & -1 \end{pmatrix}## I found the characteristic equation: ##( λ - 3 )( λ + 2) \therefore λ = 3,-2##Going back we multiply $$\begin{pmatrix} 2 - \lambda & 2\\ 2 & -1 - \lambda \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ Which gives \begin{matrix} 2x -...
  24. The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses

    The applications of eigenvectors and eigenvalues | That thing you heard in Endgame has other uses

    Zach Star gives an explanation of Eigenvalues and Eigenvectors, with some applications
  25. M

    Normalisation of eigenvectors convention for exponentiating matrices

    Hi, I just have a quick question when I was working through a linear algebra homework problem. We are given a matrix A = \begin{pmatrix} 2 & -2 \\ 1 & -1 \end{pmatrix} and are asked to compute e^{A} . In earlier parts of the question, we prove the identities A = V \Lambda V^{-1} and e^{A}...
  26. S

    Setting Free variables when finding eigenvectors

    upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...
  27. J

    Modeling the populations of foxes and rabbits given a baseline

    From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...
  28. S

    What's the formula? - eigenvectors from eigenvalues

    Anyone know what result this article is talking about? https://www.theatlantic.com/science/archive/2019/11/neutrino-oscillations-lead-striking-mathematical-discovery/602128/
  29. Greg

    MHB What is the Unexpected Discovery in Basic Math?

    A link to an interesting article I found is below: https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/
  30. entropy1

    I Orthogonal eigenvectors and measurement

    An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm...
  31. Wrichik Basu

    B Gram-Schimidt orthonormalization for three eigenvectors

    Say I have a matrix ##A## and it has three eigenvectors ##|\psi_1\rangle##, ##|\psi_2\rangle## and ##|\psi_3\rangle##. I want to orthogonalize these. Say my orthogonalized eigenvectors are ##|\phi_1\rangle##, ##|\phi_2\rangle## and ##|\phi_3\rangle##. $$ \begin{eqnarray} |\phi_1\rangle =...
  32. S

    Understanding Eigenvectors: Solving for Eigenvalues and Corresponding Vectors

    Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!
  33. Creedence

    I Eigenvectors of the EM stress-energy tensor

    My question is that what is the physical meaning of the EM stress-energy tensor's eigenvectors? Thanks for the answers - Robert
  34. C

    I Differences between the PCA function and Karhunen-Loève expansion

    Hello everyone. I am currently using the pca function from MATLAB on a gaussian process. Matlab's pca offers three results. Coeff, Score and Latent. Latent are the eigenvalues of the covariance matrix, Coeff are the eigenvectors of said matrix and Score are the representation of the original...
  35. Sunny Singh

    I Pauli matrices and shared eigenvectors

    We know that S2 commutes with Sz and so they share their eigenspace. Now since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx. But since the paulic matrices σx and σy are not diagonlized in the eigenbasis of S2, it is clear...
  36. R

    Find eigenvalues & eigenvectors

    Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at...
  37. Mutatis

    Find the eigenvalues and eigenvectors

    Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix} $$ Homework Equations Characteristic polynomial: $$ \Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...
  38. 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...
  39. Z

    Normalization & value of Eigenvectors

    Homework Statement I have got the following matrix. I have found the eigen values but in some eq x, y & z terms are vanishing, so how to find the value of eigen vector? Also why we have to do normalization?? A__=__[1__1__0] ______[1__1__0] ______[0__0__1]Homework Equations A-λI=0 Ax = -λIx...
  40. Z

    Calculating Eigenvectors: 3*3 w/o Augmented Matrix

    Homework Statement I am continuing from : https://www.physicsforums.com/threads/finding-eigen-values-list-of-possible-solutions-for-lambda.955164/ I have got a 3 * 3 matrix. I have to find itseigen values and eigen vectors. I have found the eigen values.For calculating eigen vectors they are...
  41. Z

    How Do You Find Eigenvectors of a 2x2 Matrix?

    Homework Statement Consider the following Matrix: Row1 = 2 2 Row2 = 5 -1 Find its Eigen Vectors Homework Equations Ax = λx & det(A − λI)= 0. The Attempt at a Solution First find the det(A − λI)= 0. which gives a quadratic eq. roots are λ1 = -3 and λ2 = 4 (Eigen values) Then using λ1, I...
  42. 1

    Find Eigenvectors for σ⋅n: Solving the Equations

    1. n = sinθcosφ i + sinθsinφ j + cos k σ = σx i + σy j + σz k , where σi is a Pauli spin matrix Find the eigen vectors for the operator σ⋅n 2. Determinant of (σ⋅n - λI), where I is the identity matrix, needs to equal zero (σ⋅n - λI)v = 0, where v is an eigen vector, and 0 is the zero vector...
  43. snoopies622

    B How does matrix non-commutivity relate to eigenvectors?

    Given matrices A,B and Condition 1: AB does not equal BA Condition 2: A and B do not have common eigenvectors are these two conditions equivalent? If not, exactly how are they related? Since I'm thinking about quantum mechanics I'm wondering specifically about Hermitian matrices, but I'm...
  44. M

    Calculating Eigenvectors for a 3x3 Matrix: Understanding the Process

    Hi, I am trying to find the eigenvectors for the following 3x3 matrix and are having trouble with it. The matrix is (I have a ; since I can't have a space between each column. Sorry): [20 ; -10 ; 0] [-10 ; 30 ; 0] [0 ; 0 ; 40] I’ve already...
  45. 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...
  46. P

    How Do Eigenvalues and Eigenvectors Change for Matrix B = exp(3A) + 5I?

    Homework Statement Find the eigenvalues and eigenvectors of the matrix ##A=\matrix{{2, 0, -1}\\{0, 2, -1}\\{-1, -1, 3} }## What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix?Homework EquationsThe Attempt at a Solution So I've found the...
  47. 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...
  48. S

    A Eigenvectors and matrix inner product

    Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula : \begin{equation} \int x(t)\overline y(t) dt \end{equation} on the x and y coordinates of the eigenvectors [x_1,y_1] and...
  49. F

    I Bases, operators and eigenvectors

    Hello, In the case of 2D vector spaces, every vector member of the vector space can be expressed as a linear combination of two independent vectors which together form a basis. There are infinitely many possible and valid bases, each containing two independent vectors (not necessarily...
  50. astrocytosis

    Eigenvalues and eigenvectors of a Hamiltonian

    Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...