Eigenvalue Definition and 402 Threads

Homework Statement The system described by the Hamiltonian H_0 has just two orthogonal energy eigenstates, |1> and |2> , with <1|1>=1 , <1|2> =0 and <2|2>=1 . The two eignestates have the same eigenvalue , E_0: H_0|i>=E_0|i>, for i=1 and 2. Now suppose the Hamiltonian for the...

Consider the following linear homogeneous ordinary differential equation system: (NB this system describes the movement of the natural response of a two degree of freedom structural system made up of two lumped masses connected by elastic rigidities) : \left( \begin{array}{cc}...

Suppose P: V->V s.t. P^2 = P and V = kerP + ImP (actually not just + but a direct sum). Find all eigenvalues of P. ---- Which of the following explanations is right? (1 is an eigenvalue, but is 0 also?) Could somebody please explain? ----- First answer: Suppose that λ is an...

Homework Statement Let A = a b c d A characteristic value of A (often called an eigenvalue) is denoted by λ and satisfies the relation det(A - λI) = 0 Obtain the characteristics values of E = 1 -1 -1 1 Homework Equations Well I is the unit or identity matrix 1 0 0 1...

Homework Statement Solve the system. dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2]Homework Equations The Attempt at a Solution I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors: (1-λ)(-7-λ)+16=0 λ2+6λ+9=0 λ=-3,-3 So...

Homework Statement solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9] Homework Equations The Attempt at a Solution I know I need to find the Eigenvalues but then I get a little confused from there. (λ-3)(λ+3)=0 λ=3, -3

Homework Statement Given matrix A: a 1 1 ... 1 1 a 1 ... 1 1 1 a ... 1 .. . .. ... 1 1 1 1 ... a Show there is an eigenvalue of A whose geometric multiplicity is n-1. Express its value in terms of a. Homework Equations general eigenvalue/vector equations The Attempt at a Solution My...

Hi guys, probably that's the wrong forum, but I was just curious about the plot (Figure 1 Chapter XI A./1. page 1097 / Volume II) of the eigenvalues E(\lambda). If I calculate them they are supposed to be straight lines with positive or negative slope i.e.: E(\lambda) = E_n^0 + \lambda...

I'm trying to teach myself quantum mechanics using a book I got. I made an attempt at one of the questions but there are no solutions or worked examples so I'm wondering if I got it right. Here it goes Homework Statement Suppose an observable quantity corresponds to the operator \hat{B}=...

I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me. "Properties of the eigenvalue when A is symmetric. If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times), orthogonal...

Homework Statement Prove that similar square matricies have the same eigenvalues with the same algebraic multiplicities. Homework Equations C^-1PC=Q The Attempt at a Solution Am I supposed to show that (P-\lambdaI)x=(C^-1PC-\lambdaI)x?

Homework Statement Let A be an nxn matrix and let I be the nxn identity matrix. Compare the eigenvectors and eigenvalues of A with those of A+rI for a scalar r. Homework Equations The Attempt at a Solution I think I should be doing something like this: det(A-\lambdaI), and...

I want to write myself a algorithm to solve generalised eigenvalue problems in quantum mechanics.I know there are a lot of library there that allow me to use it directly but i just want to write my own so that i can learn the mathematics methods that solve the problem... I don't know how to...

Homework Statement Prove that if two linear operators A and B commute and have non-degenerate eigenvalues then the two operators have common eigenfunctions. Homework Equations [A,B]= AB - BA= 0 Af=af Bg=cg,\ let\ g=(f+1) --> B(f+1)=c(f+1)\ where\ a\neq c The Attempt at a...

Homework Statement Show that A and AT share the same eigenvalue. Homework Equations The Attempt at a Solution let v be the eigenvector Av=Icv since ATv=ITcv and IT=I, ATv=Icv so ATv=Icv=Av so A and AT must have the same eigenvalue.

Homework Statement Let U be a fixed nxn matrix and consider the operator T: Msub(n,n)------>Msub(n,n) given by T(A)=UA. Show that c is an eigenvalue of T if and only if it is an eigenvalue of U. Homework Equations The Attempt at a Solution If T(A)=UA then T(A)-UA=0 (T-U)A=0. Let...

Another proof... Homework Statement Suppose c is an eigenvalue of a square matrix A with eigenvector X=/=0. Show that p(c) is an eigenvalue of p(A) for any nonzero polynomial p(x). Homework Equations The Attempt at a Solution Knowing that c is an eigenvalue of A, it is true that...

Hi, Is there any solution for the following problem: Ax = \lambda x + b Here x seems to be an eigenvector of A but with an extra translation vector b. I cannot say whether b is parallel to x (b = cx). Thank you in advance for your help... Birkan

Hello, I was reading something in my text/wikipedia, and they both said that "...the eigenvalues of a matrix are the zeros of its characteristic polynomial." Do they mean that λ in the characteristic polynomial causes det (A - λI) = 0 (in particular A = λI)? JL

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