What is Eigenvalues: Definition and 849 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. D

    Finding Eigenvalues to Prove trace P is Nonnegative Integer

    I'm having trouble with this: Prove that if P is a linear map from V to V and satisfies P^2 = P, then trace P is a nonnegative integer. I know if I find the eignevalues , their sum equals trace P. But how do I find them here? any thoughts? Thanks
  2. S

    Why do antisymmetric eigenvalues have to be purely imaginary?

    Why do they have to purely imaginary? I got a proof that looks like Ax=ax where a = eigenvalue therefore Ax.x = ax.x = a|x|^2 Ax.x = x.(A^t)x where A^t = transpose = -A x.(-A)x = -b|x|^2 therefore a=-b, where b = conjugate of a Now is this as far as i need to go?
  3. M

    Matrix with only real eigenvalues

    Please help me I am hopelessly lost and don't even know where to start! I guess they're right when they said girls suck at math! It's not fair! :redface: Let A be an nxn matrix with only real eigenvalues. Prove that there is an orthogonal matrix Q such that (Q^T)AQ is upper triangular with...
  4. U

    Find Eigenspace for Matrix A with 2 Eigenvalues

    matrix A = \left(\begin{array}{ccc}3&0&0 \\ 0&3&0 \\3&0&0 \end{array}\right) has two real eigenvalues lambda_1=3 of multiplicity 2, and lambda_2=0 of multiplicity 1. find the eigenspace. A = \left(\begin{array}{ccc}3-3 &0&0 \\ 0&3-3&0 \\3&0&0-3 \end{array}\right) A =...
  5. D

    Boundary value problem involving eigenvalues

    I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need...
  6. J

    Adiabatic evolution of eigenvalues

    If I have two positive definite Hermitian NxN matrices A and B, if I adiabatically change the components of A to B (constraining any intermediate matrices to be Hermitian as well, but not necessarily positive definite) while \"following\" the eigenvalues ... will the mapping of the eigenvalues...
  7. T

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  8. N

    Eigenvalues: Real & Equal in Size but Opposite Signs

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  9. Repetit

    Where Can I Learn to Code a Function for Calculating Eigenvalues?

    Hey! Does someone know of some resources which describe how to code a function which calculates the eigenvalues of a matrix? This could be either resources on the net or a book. If you know of a good book which teaches about programming and mathematics together in general I'd be happy to know...
  10. S

    Proving Nilpotency and Nonnegativity of Eigenvalues of Symmetric Matrices

    Show that every eigenvalue of A is zero iff A is nilpotent (A^k = 0 for k>=1) i m having trouble with going from right to left (left to right i got) we know that det A = product of the eignevalues = 0 when we solve for the eigenvalues and put hte characteristic polynomial = 0 then det...
  11. T

    Finding eigenvalues and eigenvectors of a matrix

    I'm asked to find the eigenvalues and eigenvectors of an nxn matrix. Up until now I thought eigenvectors and eigenvalues are something that's related to linear transformations. The said matrix is not one of any linear transformation. What do I do?
  12. fargoth

    Finding eigenvalues and eigenvectors

    is there any trick for finding the eigenvalues and vectors for this kind of matrix? \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & \sqrt{\frac{3}{2} & 0 & 0 \\ 0 & \sqrt{\frac{3}{2} & 0 & \sqrt{\frac{3}{2} & 0 \\ 0 & 0 & \sqrt{\frac{3}{2} & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array}...
  13. B

    Fredholm Integral of Second Kind, Eigenvalues

    I need help with an integral eigenvalue equation...I am lost on how to handle this: \int_{-\infty}^{\infty} dy K(x,y) \psi_n(y) = \lambda_n \psi_n(x) The kernel, K(x,y) is a 2D, correlated Gaussian. I have read that for this case an analytic solution exist for the eigenvalues, \lambda_n...
  14. G

    Can a theorem simplify finding eigenvalues of a block matrix?

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  15. Μ

    Defective eigenvalues?

    if you have a differential equation of the form x' = Ax where A is the coefficient matrix, and you get a triple eigenvalue with a defect of 1. (meaning you get v1 and v2 as the associated eigenvector). How do you get v3 and how do you set up the solutions? I tried finding v3 such that...
  16. J

    Distinct Eigenvalues and Eigenvectors in Matrix Multiplication

    Let A be an nxn mx with n distinct eigenvalues and let B be an nxn mx with AB=BA. if X is an eigenvector of A, show that BX is zero or is an eigenvector of A with the same eigenvalue. Conclude that X is also an eigenvector of B. I could show BX is zero or is an eigenvector of A with the...
  17. M

    Solving Complex Eigenvalues: Geometric Interpretation

    I've got a homework problem that I am needing to do; however, I am not sure really what the question is asking. Obviously since I don't know what is being asked, I don't know where to begin. I was hoping for some insight. Question: Show that matrix A = {cos (theta) sin (theta), -sin...
  18. M

    Eigenvectors, Eigenvalues and Idempotent

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  19. P

    Unraveling the Mysteries of Power and Inverse Power Methods for Eigenvalues

    I have two excercises which have been causing me to tear my hair off for some time now. (a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k) (b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A (c) the smallest/largest...
  20. B

    Angular momentum eigenvalues

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  21. E

    Finding Eigenvalues for Tridiagonal Matrix - Showing Orthogonality

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  22. B

    Eigenvalues of a 2 by 2 matrix

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  23. M

    Confused on finding Eigenvalues and Eigenvectors

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  24. B

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  25. B

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  26. O

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  27. F

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  28. C

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  29. J

    Eigenvalues, eigenvectors question

    1). suppose that y1, y2, y3 are the eigenvalues of a 3 by 3 matrix A, and suppose that u1, u2,u3 are corresponding eigenvectors. Prove that if { u1, u2, u3 } is a linearly independent set and if p(t) is the characteristic polynomial for A, then p(A) is the zero matrix. I thought...
  30. J

    Linear algebra: determinants and eigenvalues

    i'm reading and doing some work in introduction to linear algebra fifth edition, and i came across some problems that i had no clue. 1. An (n x n) matrix A is a skew symmetric (A(transposed) = -A). Argue that an (n x n) skew-symmetrix matrix is singular when n is an odd integer. 2. Prove...
  31. H

    Find Eigenvalues: Physics Forums Homework Help

    This thread, https://www.physicsforums.com/showthread.php?t=74810, was orignally posted here in the QM forum, but it was moved to the homework section, which is reasonable. But nobody there knows quantum mechanics. I guess the OP gave up on it, but I'm curious how to do the problem now. So if...
  32. S

    A QM problem on finding eigenvalues

    Hi. I have this problem which i am stuck at: Consider a one-dimensional Hamilton operator of the form H = \frac{P^2}{2M} - |v\rangle V \langle v| where the potential strength V is a postive constant and |v \rangle\langle v| is a normalised projector, \langle v|v \rangle = 1 ...
  33. S

    Eigenvalues of the square of an operator

    If L^2 |f> = k^2 |f>, where L is a linear operator, |f> is a function, and k is a scalar, does that mean that L|f> = +/- k |f>? How would you prove this?
  34. G

    Eigenvalues Redux: Deriving the Product of Eigenvalues = Determinant

    In a recent thread https://www.physicsforums.com/showthread.php?t=67366 matt and cronxeh seemed to imply that we should all know that the product of the eigenvalues of a matrix equals its determinant. I don't remember hearing that very useful fact when I took linear algebra (except in the...
  35. T

    4X4 real matrix eigenvalues

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  36. B

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    I having trouble finding the eigenvalues and eigenfunctions for the operator \hat{Q} = \frac{d^2}{d\phi^2}, where \phi is the azimuthal angle. The eigenfunctions are periodical, f(\phi) = f(\phi + 2\pi), which I think should put some restrictions on the eigenvalues. I think...
  37. H

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    How can i find the eigen value(s) of A - (alpha)I where A is an arbitrary matrix ?
  38. cepheid

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  39. T

    Eigenvalues Problem: Show Inverse of Diagonalizable Matrix A

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  40. C

    Solve X'+2X'+(\lambda-\alpha)X=0 | Chris Struggling

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  41. M

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  42. L

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    I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used. The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly. At that point...
  43. F

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  44. S

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  45. P

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  46. I

    How Do You Calculate Eigenvalues for a 2x2 Symmetric Matrix?

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  47. dduardo

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    I'm currently taking linear algebra and it has to be the worst math class EVER. It is extremely easy, but I find the lack of application discouraging. I really want to understand how the concepts arose and not simple memorize an algorithm to solve mindless operations, which are tedious. My...
  48. M

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  49. lethe

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    the signature of a metric is often defined to be the number of positive eigenvalues minus negative eigenvalues of the metric. this definition has always seemed a little suspicious to me. eigenvalues are defined for endomorphisms of a linear space, whereas the metric is a bilinear functional...
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