Eigenvalues Definition and 853 Threads

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. A

    Are Eigenvalues of a Non-Hermitian Matrix Real?

    Given a 4x4 non-Hermitian matrix, is there any method I can use to prove the eigenvalues are real, aside from actually computing them? I'm looking for something like the converse of the statement "M is Hermitian implies M has real eigenvalues". When can one say that the eigenvalues of a...
  2. Z

    Eigenvalues and Eigenvectors uniquely define a matrix

    Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix M from a matrix of its eigenvectors as columns P and a diagonal matrix of the eigenvalues E through M=P E P^{\dagger}?
  3. F

    What are the eigenvalues and eigenvectors of matrix A = [2 2; 3 1]?

    Eigenvalues & Eigenvectors !SOLVED! Homework Statement Find the eigenvalues and eigenvectors of matrix A = \left( \begin{array}{cc} 2 & 2 \\ 3 & 1 \end{array} \right) Homework Equations Ax = \lambda x The Attempt at a Solution Solving \left\vert \begin{array}{cc} 2 - \lambda &...
  4. B

    Can an Invertible Matrix Have Zero as an Eigenvalue?

    Homework Statement Let B be an invertible matrix a.) Verify that B cannot have zero as an eigenvalue. b.) Verify that if \lambda is an eigenvalue of B, then \lambda^{-1}^ is an eigenvalue of B^{-1}. Homework Equations Bv = \lambdav, where v\neq0The Attempt at a Solution a.) I'm pretty sure...
  5. malawi_glenn

    Eigenvalues and eigenvectors of symmetric 2x2 matrix?

    Hello I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0. I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me? cheers
  6. 9

    Eigenvalues and Eigenvectors of 3x3 matricies

    Hello Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues. Can someone please help.. Thanks
  7. J

    What are the Eigenvalues and Eigenvectors of Similar Matrices

    Homework Statement Let A and B be similar matrices a)Prove that A and B have the same eigenvalues Homework Equations None The Attempt at a Solution Firstly, i don't see how this can even be possible unless the matrices are exactly the same :S
  8. J

    Understanding Eigenvalues and Determinants with Repeated Multiplicities

    Homework Statement Let A be an nxn matrix, and suppose A has n real eigenvalues lambda_1, ...lambda_n repeated according to multiplicities. Prove that det A = lambda_1...lambda_n Homework Equations None The Attempt at a Solution Could someone explain what is meant by 'repeated...
  9. D

    Proving detA = λ1...λn for Real Eigenvalues

    Homework Statement Let A be nxn matrix, suppose n has real eigenvalues,λ1,...,λn, repeated according to multipilicities. Prove that detA = λ1...λn. Homework Equations The Attempt at a Solution I started by applying the definition, Av = λv, where v is an eigenvector. then I just dun...
  10. G

    Proving the Diagonalization of a Real Matrix with Distinct Eigenvalues

    The real matrix A= \begin{pmatrix}\alpha & \beta \\ 1 & 0 \end{pmatrix} has distinct eigenvalues \lambda1 and \lambda2. If P= \begin{pmatrix}\lambda1 & \lambda2 \\ 1 & 0 \end{pmatrix}...
  11. P

    Finding eigenvalues of a Hamiltonian involving Sz, Sz^2 and Sx

    I have the Hamiltonian for an S=5/2 particle given by: H= a.Sz + b.Sz^2 +c.Sx where Sz and Sx are the spins in z and x directions respectively. The resulting matrix is tridiagonal symmetric but I can't find the eigenvalues..Any idea how to diagonalise it. N.B: a is a variable and must be...
  12. J

    Finding eigenvalues and eigenfunctions

    Homework Statement Given X''(x) + lambda*X(x) = 0 X(0) = X'(0), X(pi) = X'(pi) Find all eigenvalues and eigenfunctions. Homework Equations Case lambda = 0 Case lambda > 0 Case lambda < 0 The Attempt at a Solution First case, X(x) = Ax + B but the function doesn't satisfy...
  13. E

    Systems Of Linear D.E's, Complex Eigenvalues

    1. Find the General Solution of the given system [ -1 -1 2 ] X = X' [ -1 1 0 ] [ -1 0 1 ] det(A-lambda*Identity matrix) = 0, solve for eigenvalues/values of lambda (A-lambda*Identity matrix|0) The eigenvalues we got are 1 and 1 +/- i. The matrix generated for...
  14. D

    Eigenvalues for an Invertible Matrix

    Homework Statement A is an invertible matrix, x is an eigenvector for A with an eiganvalue \lambda \neq0 Show that x is an eigenvector for A^-1 with eigenvalue \lambda^-1 Homework Equations Ax=\lambdax (A - I)x The Attempt at a Solution I know that I need to find x and then apply...
  15. K

    How Are Eigenvalues Used in Real Life?

    I am trying to get some intuition for Eigenvalues/Eigenvectors. One real-life application appears to be a representation of resonance. What are some practical uses for Eigenvalues? What other things may Eigenvalues represent?
  16. M

    Matrix diagonalisation with complex eigenvalues

    Homework Statement Is there a basis of R4 consisting of eigenvectors for A matrix? If so, is the matrix A diagonalisable? Diagonalise A, if this is possible. If A is not diagonalisable because some eigenvalues are complex, then find a 'block' diagonalisation of A, involving a 2 × 2 block...
  17. Y

    Symmetric matrix with eigenvalues

    Homework Statement Let {u1, u2,...,un} be an orthonormal basis for Rn and let A be a linear combination of the rank 1 matrices u1u1T, u2u2T,...,ununT. If A = c1u1u1T + c2u2u2T + ... + cnununT show that A is a symmetric matrix with eigenvalues c1, c2,..., cn and that ui is an eigenvector...
  18. L

    Eigenvalues of linear operators

    Let V be the vector space of all real integrable functions on [0,1] with inner product <f,g>=\int_0^1 f(t)g(t)dt Three linear operators defined on this space are A=d/dt and B=t and C=1 so that Af=df/dt and Bf=tf and Cf=f I need to find the eigenvalues of these operators: For A...
  19. K

    Tips on finding the eigenvalues of a 3x3 matrix

    I find it rather tedious to calculate the eigenvalues of a 3x3 matrix. For example The \emph{characteristic polynomial} $\chi(\lambda)$ of the 3$3 \times 3$~matrix \[ \left( \begin{array}{ccc} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array} \right)\] is given by the formula \[...
  20. O

    When Should I Study Thermal Physics Relative to Quantum Mechanics?

    can somebody explain eigenvalues inside hilbert spaces??
  21. R

    Upper and Lower bound eigenvalues Sturm Liouville problem

    I have 2 questions that need to be solve: 01. Find upper and lower bound for the k-th eigenvalue \lambda_{k} of the problem ((1+x^2)u')'-xu+\lambda(1+x^2)u for 0< x< 1 with boundary conditions u(0)=0 and u(1)=0 02. Find a lower bound for the lowest eigenvalue of the problem...
  22. E

    Onstruct a 3x3 matrix A that has eigenvalues

    2. Construct a 3x3 matrix A that has eigenvalues 1, 2, and 4 with the associated eigenvectors [1 1 2]T, [2 1 -2]T and [2 2 1]T, respectively. [Hint: use P-1AP = K, where K is the diagonal matrix] hlp me... pls guild me to the step reli no idea how to do it
  23. L

    What Does a Strong Eigenvalue Signify in a System of Equations?

    In a system of equations with several eigenvalues, what does it mean (signify) when one is strong (high in value) and the others are weak (low in value)? Can a general statement be made without referencing an application? If so, is there a math book that explains the idea?
  24. K

    Are All λ Smaller Than 1/4 Eigenvalues for the Given Differential Equation?

    hi I have the following eigenvalue problem -(x2y')'=λy for 1<x<2 y(1)=y(2)=0 I tried plugging an equation y=xa and you get the equation a2+a+λ=0 so for this I get that λ<1/4 to hava a solution. So does this mean, every λ smaller than 1/4 is an eigenvalue? do you know what else I...
  25. K

    Eigenvalues of Laplace eq in the circle.

    Does anybody know a web page or a book, or the general method to find the eigenvalues and the eigenfunctions of laplacian u =lambda u inside the circle u=0 in the boundary thanks
  26. C

    Eigenvalues of an Invertible Matrix

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  27. S

    Learn How to Find Eigenvalues of 3x3 Matrices | Eigenvalue Algorithm Explained

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  28. J

    Normalize both ψ1 and ψ2 and then find the energy eigenvalues of each

    Hi there this is my first post here, I am having some trouble with a homework question in quantum. I need to normalize both ψ1 and ψ2 and then find the energy eigenvalues of each. The given Hamiltonian is H0 = (1 2 ) (2 -1) And ψ1 = (...
  29. S

    About an equation with eigenvalues

    Let's say I have the equation p(t)f''(t)=Kf(t) with p(t) a known periodical function, K an unknown constant and f(t) the unknown function. This is an eigenvalues problem that once solved gives a set of K={k1, k2,...} eigenvalues. I get these eigenvalues and they coincide with the ones...
  30. L

    Computer Vision, Corners and Eigenvalues

    This question is about the use of eigenvalues in a specific application. The subject is Computer Vision and the topic is the Harris Corner detection method. The attached file is PDF document of slides that show the math in a bit more detail. In the slides, a corner is located by looking at...
  31. C

    Relationship between determinant and eigenvalues?

    Homework Statement Find the eigenvalues of B = [5 2 0 2], [3 2 1 0], [3 1 -2 4], [2 4 -1 2]. Compute the sum and product of eigenvalues and compare it with the trace and determinant of the matrix. Homework Equations The Attempt at a Solution I get the characteristic polynomial...
  32. P

    What do the directions of eigenvalues represent?

    Background: I'm having trouble using principal component analysis to try and align two data sets. I have two sets of 3D point data, and I can use PCA to get principal axes of the two sets of data. I do this by finding the eigenvectors of the covariance matrix for each set of data. This gives...
  33. C

    Can You Simplify Finding Eigenvalues of an n x n Matrix?

    Just wondering is there a way to get the characteristic equation of an n by n matrix without going through tedious calculations of solving multiple determinants of matrices?
  34. E

    The lowest energy eigenvalues

    Homework Statement compute and plot the 10 lowest energy eigenvalues of a particleinan infinity deep spherically symmetric square well? Homework Equations The Attempt at a Solution
  35. E

    Plot a several energy eigenvalues of a partical

    Hi How can i compute(or obtain from mathmatical tables) and plot a several energy eigenvalues of a particle in an infinity deep ,spherically and symmetric square well?
  36. N

    Linear Systems of ODE's: Eigenvalues and Stability

    Homework Statement Hi all. I am given by following linear system: \begin{array}{l} \dot x = dx/dt = ax \\ \dot y = dy/dt = - y \\ \end{array} The eigenvalue of the matrix of this system determines the stability of the fixpoint (0,0): A=\left( {\begin{array}{*{20}c} a & 0 \\...
  37. M

    Eigenvalues with a leslie population model

    Homework Statement In the Leslie population model, suppose matrix A has a strictly dominant eigenvalue \lambda_1. Age class evolution is given by: x^{(k)} = Ax^{(k-1)}; initial population is x^{(0)}. (i) Initially, let x^{(0)} be the linear combination a_1x_1 + a_2x_2 + ... + a_nx_n, of A's...
  38. J

    Eigenvalues - real and imaginary

    Am I understanding this right? Let's say I have a 15x15 matrix called Z. Then the matrix of eigenvalues calculated from Z, called D, can have two forms - either diagonal or block diagonal. If the matrix D comes out with values only on the diagonal, then there are only real values. But...
  39. S

    Are Eigenvalues and Eigenvectors Correctly Understood in Matrix Operations?

    Homework Statement I am studying about eigenvalues and norms. I was wondering whether the way I understand them is correct. Homework Equations The Attempt at a Solution The eigenvalue of a matrix those that satisfy Ax = \lambda x, where A is a matrix, x is an eigenvector, \lambda...
  40. F

    Hermitian Operators and Eigenvalues

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  41. R

    Problems related to eigenfunctions and eigenvalues

    can somebody help me with the solution of the following problems? Ques. Find the eigenfunctions and eigenvalues for the operators 1. sin d/d psi 2. cos(i d/d psi) 3. exp(i a d/d psi) 4. (d)square/d (x)square+z/x * d/dx
  42. Q

    Product eigenstate eigenvalues

    Homework Statement What are the eigenvalues of the set of operators (L1^2, L1z, L2^2, L2z) corresponding to the product eigenstate \left\langlem1 l1 | m2 l2 \right\rangle? PS: If you have Liboff's quantum book, this is problem #9.30. Homework Equations We've also been learning...
  43. J

    Connection between isolated eigenvalues and normalizable eigenstates

    It seems to be true, that if some eigenvalue of a Hamilton's operator is an isolated eigenvalue (part of discrete spectrum, not of continuous spectrum), then the corresponding eigenstate is normalizable, and on the other hand, if some eigenvalue of a Hamilton's operator is not isolated, then the...
  44. E

    Diagonalizable Matrices & Eigenvalues

    Hello, Is it sufficient to determine that a nXn matrix is not diagonalizable by showing that the number of its distinct eigenvalues is less than n? Thanks for your time.
  45. V

    Question regarding eigenvalues of angular momentum operator

    Hello! First of all let me wish you a happy new year! This is not a homework problem, but rather a curiosity of mine. In Schwabl's Quantum Mechanics, one can find the proof of the fact that all eigenvalues of the angular momentum Lz are either integers or half-integers, raging from -l to l(l...
  46. F

    Why Is (A-3I)^2 Used Instead of (A-3I)^3 in Finding Eigenvectors?

    my question is take A= {(5,0,-1),(2,3,-1),(4,0,1)} find all eigenvalues and eigenvectors by using the characteristic equation i get -(lamda-3)3 however its the next bit i don't understand, in the answers (A-3I)(x,y,z)=(0,0,0) is used which I'm perfectly ok with and then (A-3I)2 is used and...
  47. F

    Solving Eigenvalues of Hessian Matrix

    g(x,y) = x^3 - 3x^2 + 5xy -7y^2 Hessian Matrix = 6x-6******5 5********-7 Now I have to find the eigenvalues of this matrix, so I end up with the equation (where a = lambda) (6x - 6 - a)(-7 - a) - 25 = 0 Multiplying out I get: a^2 - 6xa + 13a - 42x + 17 = 0 How am I supposed to solve...
  48. J

    Matrix, eigenvalues and diagonalization

    Matrix A= 1 2 0 2 1 0 2 -1 3 i got eigenvalues k=3 k=-1 what do i do after that to prove it is not able to be diagonalized
  49. J

    Is There an Easier Way to Find the Signs of Eigenvalues for Sparse Matrices?

    Homework Statement Hey guys, for my linear algebra class I need to find the signs of the eigenvalues (I just need to know how many are positive and how many are negative) of an nxn matrix with zeros everywhere except for the two diagonals directly above and directly below the main diagonal...
  50. S

    What are the eigenvalues of a non-Hermitian operator?

    Hi, everyone! While I was studying for my midterm, I encountered this question. ------ Consider the hermitian operator H that has the property that H4 = 1 What are the eigenvalues of the operator H? What are the eigenvalues if H is not restricted to being Hermitian? ------ What I am...
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