Eigenvalue Definition and 382 Threads

Theorem: Let A be in M_n_x_n(F). Then a scalar \lambda is an eigenvalue of A if and only if det(A - \lambda I_n) = 0. Proof: A scalar lambda is an eigenvalue of A if and only if there exists a nonzero vector v in F^n such that lambda*v, that is (A - \lambda I_n)(v) = 0. By theorem 2.5, this...

Homework Statement I would like to know what the definition of a Differential Eigenvalue Problem is please? I am a maths undergraduate. Homework Equations \lambda y = L y, where \lambda is eigenvalue, L is a linear operator. The Attempt at a Solution I have searched via google...

Let x not equal to zero be a vector in the nullspace of A. Then x is an eigenvector of A. I'm not sure how to start this proof

Homework Statement Let x be a unit vector. Namely x(Transpose)*x = 1. If (A − Let x be a unit vector. If (A − λI)x = b, then λ is an eigenvalue of A − bx(transpose). The Attempt at a Solution I have no idea where to start this proof.

Homework Statement Let λ be an eigenvalue of A. Then λ^2 is an eigenvalue of A^2 The Attempt at a Solution I know I have to start by using the fact that λ is an e.v of A then set up an equation relating the eigenvalues and vectors to A which is: Ax=λx. And I understand that the...

Homework Statement "Let A be a diagonalizable n by n matrix. Show that if the multiplicity of an eigenvalue lambda is n, then A = lambda i" Homework Equations The Attempt at a Solution I had no idea where to start.

Homework Statement Let λ be an eigenvalue of A. Then λ+σ is an eigenvalue of A+σI Homework Equations The Attempt at a Solution I'm guessing I need to use the fact that λ is an e.v of A to start with. But then when I add σ to both sides somehow I feel like I'm begging the question..

Homework Statement If λ is and eigenvalue of the the matrix A then 3λ is an eigenvalue of 3A Homework Equations The Attempt at a Solution . . . λ is an e.v of A Therefore, ∃ x not equal to 0 s.t Ax=λx Then, 3Ax=3λx which can written as 3(Ax)=3(λx)=λ(3x) and 3x does not...

Homework Statement Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalue of A Homework Equations The Attempt at a Solution since λ=2,and it is a 3*3 matrix i get the det(λI-A)=(λ-2)^3=0 then λ^3-6λ^2+12λ-8=0 now i use...

The problem is A particle of mass m and electric charges q can move only in one dimension and is subject to a harmonic force and a homogeneous electrostatic field. The Hamiltonian operator for the system is H= p2/2m +mw2/2*x2 - qεx a. solve the energy eigenvalue problem b. if the...

Homework Statement This isn't a homework problem, just something I've been trying to conceptualize for a while. Can anyone exemplify with a physical analog the concept of eigenstates? For example, I know that eigenvalues of variables with continuous spectra do not exist in the physical...

Homework Statement In Uniform Acceleration Motion, the force F is constant. then potential V(x)=Fx, and Hamiltonian H=(p^2/2m)-Fx The problem is to solve the eigenvalue problem Hpsi(x)=Epsi(x) Homework Equations F=constant V(x)=Fx H=(p^2/2m)-Fx The Attempt at a Solution I have...

Homework Statement Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A? a. [1,0] b. [0,1] c. [1,1] (The answer can be any or all of these) The Attempt at a Solution I...

Homework Statement For the matrix A = -1, 5 -2, -3 I found the eigenvalues to be -2 + 3i and -2 - 3i. Now I need some help to find the eigenvectors corresponding to each. Homework Equations The Attempt at a Solution For r = -2 + 3i, I plugged that into the (A - Ir) matrix...

Hello everybody, I have a question for which I cannot find the answer around, any help would be really appreciated. Suppose we have a matrix A of a linear transformation of a vector space, with only one eigenvalue, say 's'. My question is: Is the operator (A-sI) nilpotent? ('I' is the...

To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue. Let's say I already have some method that gives me all the eigenvectors and all...

The wave function of "particle in a box" is Asin(kx). Since potential energy is zero inside the box, so the Hamiltonian is just kinetic energy In principle, I should be able to find eigenvalue of momentum using momentum operator, but stymied in solving the equation. Can somebody help me find...

Hi everyone, I am stuck with the following for last couple of days. Many books mention during the development in the idea of Eigenvalue problem: say, you have the equation [\ A-\lambda\ I]\ X=\ 0 where A is an NxN matrix and X is an Nx1 vector. The above consists of n equations.Say,all...

Homework Statement 3.) Stress analysis at a critical point in a machine member gives the three-dimensional state of stress in MPa as the following: y = [ 105 0 0 0 -140 210 0 210 350 ]...

Hello, I am using COMSOL (RF modul) for some time now to calculate the eigenvalues (modes) of optical fibers. So far I changed all the params from hand in the software. But it is getting anoying to change a parameter (the wavelength) and to recalculate if one wants to calculate for many...

How do I prove that if A is an invertible matrix and lambda does not equal zero then one dived by lambda is an eigenvalue of the inverse of A?

Homework Statement Bessels equation of order n is given as the following: y'' + \frac{1}{x}y' + (1 - \frac{n^2}{x^2})y = 0 In a previous question I proved that Bessels equation of order n=0 has the following property: J_0'(x) = -J_1(x) Where J(x) are Bessel functions of...

Homework Statement Prove that an orthogonal transformation T in Rm has 1 as an eigenvalue if the determinant of T equals 1 and m is odd. What can you say if m is even? The attempt at a solution I know that I can write Rm as the direct sum of irreducible invariant subspaces W1, W2, ..., Ws...

Hi I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A 5 5 5 5 5 5 5 5 5 The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there...

Homework Statement solve the eigenvalue problem ∫(-∞)x dx' (ψ(x' ) x' )=λψ(x) what values of the eigenvalue λ lead to square-integrable eigenfunctions? The Attempt at a Solution ∫(-∞)xdx' (ψ(x' ) x' )=λψ(x) differentiate both sides to get ψ(x)x=λ d/dx ψ(x) ψ(x)x/λ=...

Homework Statement Find all eigenfunction of momentum operator in x(px=h/i*d/dx) and their eigenvalues. Homework Equations operator*eigenfunction=eigenvalue*eigenfunction Operator=px The Attempt at a Solution I really don't have any clues Thank you

I read from a book and claim that for any hermitian matrix can be diagonalized by a unitary matrix whose columns represent a complete set of its normalized eigenvectors. It then given an equation...

Homework Statement This is the original question: \frac{d^{2}y}{dx^{2}}-\frac{6x}{3x^{2}+1}\frac{dy}{dx}+\lambda(3x^{2}+1)^{2}y=0 (Hint: Let t=x^{3}+x) y(0)=0 y(\pi)=02. The attempt at a solution This might be all wrong, but this is all I can think of \frac{dt}{dx}=3x^{2}+1 so...

Solve the eigenvalue problem \frac{d^2 \phi}{dx^2} = -\lambda \phi subject to \phi(0) = \phi(2\pi) and \frac{d \phi}{dx} (0) = \frac{d \phi}{dx} (2 \pi). I had the solution already, but am looking for a much simpler way, if any. EDIT: Sorry that I accidentally posted...

If a vector v\in V and a linear mapping T:V\to V are fixed, and there exists numbers \lambda_1\in\mathbb{C}, n_1\in\mathbb{N} so that (T - \lambda_1)^{n_1}v = 0, is it possible that there exists some \lambda_2\neq\lambda_1, and n_2\in\mathbb{N} so that (T - \lambda_2)^{n_2}v = 0? (Here...

Why did Schrodinger call his equation eigenvalue problem? We can solve Schrodinger equation since it's just differential equation with complex number

Homework Statement V is an eigenvector of the nxn matrix A, with a eigenvalue of 4. explain why V is a eigenvector of A^2+2A+3I. what is the associated eigenvalue? Homework Equations The Attempt at a Solution is the eigenvalue of A^2+2A+3I=21?

Homework Statement Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace. T is the transformation on R2 that reflects points across some line through the origin. The Attempt at a Solution Since they tell us that...

Find a 2\times 2 matrix A for which E_4 = span [1,-1] and E_2 = span [-5, 6] where E_(lambda) is the eigenspace associated with the eigenvalue (lambda) relevant equations: Av=(lambda)v The Attempt at a Solution I've pretty much gotten most of the eigenspace/value problems down, but this...

For which value of k does the matrix A= |4 k| |-7 -5| have one real eigenvalue of multiplicity 2? The Attempt at a Solution - I tried by setting this problem up with det(A-(lambda)I) and trying to solve like that, but I can't seem to get it that way either. I am getting...

Homework Statement operator is d2/dx2 - bx2 function is psi=e^-ax2 if this fuction is eigenfuction for this operator, what is "a" and "b" constants value? Homework Equations The Attempt at a Solution

I hava a problem finding out how this is showned If A is n x n and r is not 0. Show that CrA(x) = (r^n) * CA(x/r) What rule should I think of in defanition.

Guys I read a little on how Heisenberg's quantum mechanics equations (solving with eigenvectors) were derived in the book "What is quantum mechanics: A physical adventure". There is no exercise in the book. After reading, I still don't understand eigenvalue. What is it for? How to use it...

Homework Statement Consider the following problem: if \ A \psi=\lambda\psi,prove that \ e ^ A \psi=\ e ^\lambda\psi Homework Equations The Attempt at a Solution This is my attempt.Please check if I am correct. If \ e ^ A \psi=\ e ^\lambda\psi is correct, we should...

cos a -sin a sin a cos a How do I find the eigenvalue of this rotation matrix? I did the usual way, but didn't work! Could someone tell me how to start this problem?

I know that if T has eigenvalue k, then T* has eigenvalue k bar. But if T has eigenvector x, does T* also have eigenvector x? If so, how do you prove it? I don't see that in my textbook.

Reading Sam Treiman's http://books.google.de/books?id=e7fmufgvE-kC" he nicely explains the dependencies between the Schrödinger wave equation, eigenvalues and eigenfunctions (page 86 onwards). In his notation, eigenfunctions are u:R^3\to R and the wavefunction is \Psi:R^4\to R, i.e. in contrast...

In an experiment, do we measure the eigenvalue or the expectation value ? If both can be measured, how can we distinguish one from another ?

Homework Statement The linear operator T on R^2 has the matrix [4 -5; -4 3] relative to the basis { (1,2), (0,1) } Find the eigenvalues of T. Obtain an eigenvector corresponding to each eigenvalue.Homework Equations The Attempt at a Solution I was able to find the eigenvalues (8 and -1)...

Hey, Im working with Comsol and doing some eigenvalue analysis. Why is sometimes the eigenvalues are complex numbers and not real number frequencies? How should I interpret these complex eigenvalues? Thanks

Solve the eigenvalue problem O_{6} \Psi(x) = \lambda \Psi(x) O_{6}\Psi(x) = \int from negative infinity to x of dxprime *\Psi(xprime) * xprime what values of eigenvalue \lambda lead to square integral eigenfuctions? (Hint: Differentiate both sides of the equation with respect to x) Im...

[SOLVED] Functions, operator => eigenfunction, eigenvalue Homework Statement Show, that functions f1 = A*sin(\theta)exp[i\phi] and f2 = B(3cos^{2}(\theta) - 1) A,B - constants are eigenfunctions of an operator http://img358.imageshack.us/img358/3406/98211270ob1.jpg and find...

Hello, I want to prove that the function \mathcal{A} in the 1D case satisfy \mathcal{A}=\frac{48}{m}\sum_{j=1}^\infty \frac{\sin^2(qj/2)}{j^5}=\frac{12}{m}\left[2\zeta(5)-\text{Li}_5(e^{iq})-\text{Li}_5(e^{-iq})\right], with \text{Li}_n(z) the polylogarithm function, and the matrix...

I have some question on energy eigenvalue and eigenfunction help please A particle, mass m , exists in 3 dimensions, confined in the region 0< x < 2L, 0 < y < 3L, 0 < z < 3L a) what are the energy eigenvalues and eigenfunctions of the particle? b) if the particel is a...

Hi, This is just a quick question -- I'm puzzled by the way this answer sheet represents the potential function. The question asks us to determine the energy eigenvalues of the bound states of a well where the potential drops abruptly from zero to a depth Vo at x=0, and then increases...