# What is Eigenvalues: Definition and 848 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. ### I Commutators, operators and eigenvalues

Hi I just wanted to check my understanding of something which has come up when first studying path integrals in QM. If x and px are operators then [ x , px ] = iħ but if x and px operate on states to produce eigenvalues then the eigenvalues x and px commute because they are just numbers. Is...
2. ### A Parameter optimization for the eignevalues of a matrix

Hello! I have a matrix (about 20 x 20), which corresponds to a given Hamiltonian. I would like to write an optimization code that matches the eigenvalues of this matrix to some experimentally measured energies. I wanted to use gradient descent, but that seems to not work in a straightforward...
3. ### I Tests for PSD in matrices

On a side note I'm posting on PF more frequently as I have exams coming and I need some help to understand some concepts. After my exams I will probably go inactive for a while. So I'll get to the point. Suppose we have a matrix A and I wish to check if it is positive semi definite. So one easy...
4. ### Engineering How do I find the transition matrix of this dynamic system?

Hello! I have the following matrix (picture 1.)and I am susposed to find the transition matrix ($$\phi$$) now for that I need the eigenvalue and vectors of this matrix A. The eigenvalues are 1,1 and 2. The eigenvectors I have found to be (1 0 0) (1 1 0) (5 3 1). Now to find the transition...
5. ### Model for a Qubit system using the Hamiltonion Operator

Hi, unfortunately, I am not sure if I have calculated the task a correctly. I calculated the eigenvalues with the usual formula ##\vec{0}=(H-\lambda I) \psi## and got the following results $$\lambda_1=E_1=-\sqrt{B^2+\nabla^2}$$ $$\lambda_2=E_2=\sqrt{B^2+\nabla^2}$$ I'm just not sure about...
6. ### Degenerate Perturbation: Calculating Eigenvalues

Say a model hamiltonian with unperturbed eigenvalues E1 and E2 = E3 is subjected to a perturbation V such that V12 = V21 = x and V13 = V31 = x2, with all other elements zero. I'm having trouble calculating the corrected eigenvalues. In the degenerate subspace spanned by |2> and |3> I need to...
7. ### 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!
8. ### 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...
9. ### Proving eigenvalues of a 2 x 2 square matrix

For this, Does someone please know why the equation highlighted not be true if ##(A - 2I_2)## dose not have an inverse? Many thanks!
10. ### Using inverse to find eigenvalues

For this, I don't understand how if ##(A - 2I_2)^{-1}## has an inverse then the next line is true. Many thanks!
11. ### 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...
12. ### 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!
13. ### Condition such that the symmetric matrix has only positive eigenvalues

My attempt: $$\begin{vmatrix} 1-\lambda & b\\ b & a-\lambda \end{vmatrix} =0$$ $$(1-\lambda)(a-\lambda)-b^2=0$$ $$a-\lambda-a\lambda+\lambda^2-b^2=0$$ $$\lambda^2+(-1-a)\lambda +a-b^2=0$$ The value of ##\lambda## will be positive if D < 0, so $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...
14. ### 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. ### I Velocity operator, its expression and eigenvalues

Cohen Tannoudji pp 215 pp 225 pp 223 From above we can say that there exists a velocity operator ##\mathbf v=\frac{\mathbf p}{m}## ,whose eigenvalues are the observed values of velocity. 1. I've seen multiple times that we can't define velocity in quantum mechanics, but here I find that...
16. ### A Does the Maximum Lyapunov exponent depend on the eigenvalues?

I am currently reading this paper where on page 8, the authors say that: This correlates with Figure 8 on page 12. Does it mean that there is a real correlation between eigenvalues and Lyapunov exponents?
17. ### I Getting eigenvalues of an arbitrary matrix with programming

I have learnt about the power iteration for any matrix say A. How it works is that we start with a random compatible vector v0. We define vn+1 as vn+1=( Avn)/|max(Avn)| After an arbitrary large number of iterations vn will slowly converge to the eigenvector associated with the dominant...
18. ### A Changing Hamiltonian with some eigenvalues constant

Suppose some quantum system has a Hamiltonian with explicit time dependence ##\hat{H} := \hat{H}(t)## that comes from a changing potential energy ##V(\mathbf{x},t)##. If the potential energy is changing slowly, i.e. ##\frac{\partial V}{\partial t}## is small for all ##\mathbf{x}## and ##t##...
19. ### MATLAB FEM, Matlab and the modes of an element

Hello I wrote a Matlab code to form the 8 by 8 stiffness matrix of a single, 4-noded element, for a plane strain problem for an isotropic element. I conduct an eigenvalue analysis on this matrix Matlabe reports 5 non-zero eigenvalue modes, and 3 zero-eigenvalue modes (as expected) Of the 3...
20. ### X^4 perturbative energy eigenvalues for harmonic oscillator

The book(Schaum) says the above is the solution but after two hours of tedious checking and rechecking I get 2n^2 in place or the 3n^2. Am I missing something or is this just a typo?
21. ### I Calculate Eigenvalues of Electromagnetic & Stress-Energy Tensors

How can we (as nicely as possible... i.e. not via characteristic polynomial) calculate the eigenvalues of ##F_{ab} = \partial_a A_b -\partial_b A_a## and ##T_{ab} = F_{ac} {F_b}^c- (1/4) \eta_{ab} F^2 ## and what is their physical meaning?
22. ### I Solving a Particle on the Surface of a Sphere: Obtaining Eigenvalues

The Hamiltonian of a particle of mass ##m## on the surface of a sphere of radius ##R## is ##H=\frac{L^2}{2mR^2}## where ##L## is the angular momentum operator. I want to solve the TISE ##\hat{H}\psi=E\psi## and in order to do that I rewrite ##L^2## in Schroedinger's representation in spherical...

46. ### Finding energy eigenvalues with perturbation

I know the basis I should use is |m_1,m_2> and that each m can be 1,0,-1 but how do I get the eigenvalues from this?
47. E

### Eigenvalues of an orthogonal matrix

I'm fairly stuck, I can't figure out how to start. I called the matrix ##\mathbf{A}## so then it gives us that ##\mathbf{A}\mathbf{A}^\intercal = \mathbf{I}## from the orthogonal bit. I tried 'determining' both sides... $$(\det(\mathbf{A}))^{2} = 1 \implies \det{\mathbf{A}} = \pm 1$$I don't...
48. ### I Phase Plane Diagram w/ Complex eigenvalues

Is the spiral I drew here clockwise or counterclockwise ? What’s a trick to know whether it’s going CCW or CW. Thanks!
49. ### Finding the eigenfunctions and eigenvalues associated with an operator

The eigenvalue equation is $$\frac{d^2}{d \phi^2} f(\phi) = q f(\phi)$$ This is a second order linear homogeneous differential equation. The second order polynomial associated to it is $$\lambda ^2 - q = 0 \rightarrow \lambda = \pm \sqrt{q}$$ As both roots are real and distinct, the...
50. ### 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...