Eigenvalue Definition and 402 Threads

So, my problem statement is: Suppose that two operators P and Q satisfy the commutation relation [Q,P] = Q . Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. This shouldn't be too difficult, but...

Homework Statement Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue. Homework Equations u1(q)=A*q*exp((-q^{2})/2) The Attempt at a Solution Ok, so I know that the Quantum Harmonic Oscillator...

Assuming the matrix is positive definite (necessary for cholesky decomposition). Which is faster? Which is more accurate? Is there a reliable source that has all the most common decompositions listed in order of accuracy and speed?

I've been trying to invert a real symmetric matrix and the inverse that I compute via eigenvalue decomposition is not the inverse (using QV^-1Q^T), the stranger thing is that QVQ^T gets back my orginal matrix matrix. Even more unusual is that the matrix starts off at approximately identity (in...

function: e^-(x^2/2) operator: d^2/dx^2 -x^2 The answer key says the function is an eigenfunction of the operator with an eigenvalue of -6. I can't figure out how to reach this conclusion. Also, Wolfram Alpha says d/dx(d/(dx)e^(-x^2/2)) = e^(-x^2/2) (x^2-1). Isn't this inconsistent with...

Homework Statement αo, α1,..., αd \inℝ. Show that αo + α1λ + α2λ2 + ... + αdλd \inℝ is an eigenvalue of αoI + α1A + α2A2 + ... + αdAd \inℝ^{nxn}. 2. The attempt at a solution If λ is an eigenvalue of A, then |A - Iλ| = 0. Also, λn is an eigenvalue An. So we basically have to somehow...

Proof involving nonsingular matrices. Homework Statement If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix. I've proved it like this: Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B. B-1 = (I - A)-1(I +...

Homework Statement Let A and B be symmetric matrices and X is a vector in the eigenvalue problem AX-λBX=0 a) Show that the eigenvectors are orthogonal relative to A and B. b) If the eigenvectors are orthonormal relative to B , determine C such that (C-λI)X=0, where C is a diagonal...

Hello, Given the hamiltonian : H = -( aS_z^2 + b(S_+^2 +S_-^2) ) with S=1 and a,b>0 are constants. working with the base: { |m=1> , |m=-1> , |m=0> } The matrix form of H is: H = \left( \begin{array}{ccc} -ah^2 & -bh^2 & 0 \\ -bh^2 & -ah^2 & 0 \\ 0 & 0 & 0 \end{array}...

hi i know how to calculate eigenvalue of given matrix. I want to know if two non homogenous simutaneous equation are given - than can we find its eigenvalue.

we are given B = CAC^-1 Prove that A and B have the same characteristic polynomial given a hint: explain why ƛIn = CƛInC^-1 what I did was: B = CAC^-1 BC = CA Det(BC) = Det(CA) Det(B) Det(C) = Det(C) Det(A) Now they’re just numbers so I divide both sides by Det(C) Det(B) = Det(A)...

Homework Statement Let Q be an orthogonal matrix with an eigenvalue λ_{1} = 1 and let x be an eigenvector belonging to λ_{1}. Show that x is also an eigenvector of Q^{T}. Homework Equations Qx = λx where x \neq 0 The Attempt at a Solution Qx_{1} = x_{1} for some vector x_{1}...

Homework Statement Show that the matrix A = [cos θ -sin θ sin θ cos θ] will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result. Homework Equations Ax = λx, so det(A-λI) = 0 The Attempt at a Solution In this case...

So I understand the idea of eigenvalues, eigenvectors, and eigenfunctions corresponding to a given operator on some vector or function space. But I'm just wondering, why are eigenvalues so important in quantum mechanics and physics in general? What I mean is, why are scaled multiples of a...

Homework Statement I am solving a Hamiltonian including a term \begin{equation}(x\cdot S)^2\end{equation} Homework Equations The Hamiltonian is like this form: \begin{equation} H=L\cdot S+(x\cdot S)^2 \end{equation} where L is angular momentum operator and S is spin operator. The...

Say we have an eigenvalue \lambda and corresponding eigenvectors of the form (x,x,2x)^T. What is the geometric multiplicity?

Homework Statement The problem amounts to finding the eigenvalues of the matrix |0 1 0| |0 0 1| |1 0 0| (I have no idea how to set up a matrix in the latex format, if anyone can tell me that'd be great) Homework Equations The characteristic equation for this matrix is...

Homework Statement Given the Sturm-Liouville system: y'' + λy = 0 , y(0) - y'(0) = 0 , y(1) + y'(1) = 0 Show using the Rayleigh Quotient that the eigenvalues are positive. Show that these eigenvalues are given as the solutions of the transcendental equation: tan ( √λ ) =...

I'm tinkering with a code snippet where a part finds eigenvalues. eig(A); The thing is, I tried to do it not using eig() to grasp this and got stuck. Could anyone shed some light on this..? How do I find the smallest eigenvalue?

Homework Statement An electron moves in a one-dimensional lattice with the separation between adjacent atoms being equal to a. a. Write down the momentum eigenvalue equation for the electron. b. Find the general form of the solutions of the eigenvalue equation. c. By requiring that the...

Homework Statement Proof: Prove that if A is an nxn (square mtx) such that A^2=A, then A has 0 or 1 as an eigenvalue. The Attempt at a Solution A=A^2 A^2-A=0 A(A-I)=0 A=0 or A=1 and then plugging the A solutions into the characteristic equation and solving for λ

In my lecture notes my prof used the eigenvalue c= 1 + i and ended up with the matrix with (5 3+i) as row 1, and the second row is zeroes. After that, he simply wrote that the basis for this eigenvalue c is (3+i,-5) (in column form) without explaining. How did he get that basis? I tried working...

I have a generalised eigenvalue problem of the form A\boldsymbol{u} = \lambda B\boldsymbol{u}\;, where A and B are symmetric matrices with real symbolic entries. I'm trying to compute the eigenvalues with Mathematica using the command Eigenvalues[{A,B}] which according to the documentation...

Hi guys I have a problem that I need some help with, I am looking for a direct eigenvalue solver algorithm. The problem is that all the eigenvalue solvers I can find seems to reorder the final matrix after the size of the eigenvalues. The matrix it shall calculate is very small (5-10)...

I'm trying to do something that requires solving an eigenvalue problem of the form A_{imkl} c_m c_k c^*_l=\lambda c_i where A is a known rank-4 tensor, \lambda is the eigenvalue, and the c_i's are a set of unknown coefficients that I need to determine. I would guess that this type of problem...

Q)Different eigenvectors corresponding to an eigenvalue of a matrix must be linearly dependant? Is the above statement true or false.Give reasons.

Hey there, first timer poster here Homework Statement I'm working on a barotropic linear instability analysis and I've been having trouble getting an expression for the complex phase speed eigenvalue C = C_r + i*C_i for the purpose of plotting a dispersion diagram (C_i vs k or C_r vs...

Homework Statement Consider the matrix A=[a d f; 0 b e; 0 0 c], where all elements are real numbers (a) what condition(s) on the elements of A are sufficient to guarantee that A has 3 distinct eigenvalues? (b) prove that any two eigenvectors x1 and x2 associated with two distinct...

A is a simetric metrices nxn. so v\in R^n and v\neq 0 so (\lambda I -A)^2=0 for some \lambda prove that for the same v (\lambda I -A)=0 how i tried to solve it: i just collected data from the given. simetric matrices is diagonizable. B=(\lambda I -A) we were given that B^2v=0 so...

Homework Statement This is an example from Gasiorowicz's Quantum Physics. "Example 3-1" is a particle in an infinite potential-well, but that should not matter. Homework Equations The Attempt at a Solution Why is P(-2) (which I suppose is the probability that the eigenvalue -2 is...

Okay, I know that if I can't get n linearly independent eigenvectors out of a matrix A (∈ℝnxn), it is not diagonalizable (and that some necessary conditions for diagonalizability in this regard may be being symmetric and/or having distinct eigenvalues.) This is how things are for the usual...

Dear all, in these http://pages.unibas.ch/comphys/comphys/TEACH/SS04/course.pdf" lecture notes, the author says on page (0-120): http://img15.imageshack.us/img15/615/capturena.png It is not obvious to me, why due to the translation invariance of the energy 3 eigenvalues of the D_IJ matrix have...

Homework Statement Please take a look at the attachment for the problem statement. Homework Equations For 1 dim Harmonic oscillator, E = (n+1/2)h.omega/2pi I don't know which equation to use for 2 dim The Attempt at a Solution I am unable to solve because I don't know which...

Could someone please walk me through answering this question: it looks easy but i forgot how to do it -__- i did A - lamda I = 0, substituted lamda = 5, then my matrix became 0 4 4 0 -2 -2 0 -2 -2 so i put in the form of A|b where b was a zero vector and i row reduced, getting 0 1 1 | 0 0...

Homework Statement a) |-1 1 1| | 1 -1 1| = A | 1 1 -1| Find an orthoginal matrix P that diagonalizes Ab) |0 1| What value of a is multiplicity 2, what value of a is eigen values -1 and 2 A = |a 1| what value of a does A have real eigenvaluesC) If A is a...

I have a Sturm-Liouville system \frac{d}{dx}p(x)\frac{du}{dx} - q(x)u(x)+\lambda \rho(x) u(x) = 0 with p(x) = (1-x^2)^{2p} q(x) = k^2 \rho(x) = (1-x^2)^{p-1} (p,k^2 are positive real) u(x) is defined on the interval (-A,A) where 0<A<=1. Boundary condition that u(x) satisfies is...

Hello everyone, I am solving an eigenvalue problem. Right now, I would like to know; How to determine the degeneracy of eigensolution of sturm-liouville differential eigenvalue problem? I have an eigenvalue sturm-liouville problem H f(y) = E f(y) where H is a differential operator and E is...

I have a rather basic question about solving eigenvalue problems. Once you actually find all the eigenvalues for a given operator in some basis and you go about finding the respective eigenvectors through the components and run into a situation like this: \mid \omega = 1 > \Rightarrow...

Hello everyone, please help me to answer this question. Is this true that any LTI system can be characterized by either its impulse response or engenvalue?

1. Homework Statement Suppose that A is a square matrix and the sum of the entries of each row is some number k. Is k an eigenvalue of A? if so, what is the corresponding Eigenvector?2. Homework Equations Ax-λx=0 3. The Attempt at a Solution (1-k)(K-λ)-k=0I am not sure how to solve this...

I am working on a problem and before I post the remaining questions on it, I want to be sure I calculated the eigenvector correctly. The eigenvalue I used was lambda = 3-4i. \begin{bmatrix} 3-lambda & -4\\ 4 & 3-lambda\end{bmatrix} After substituting, the eigenvector I came up with is V...

Homework Statement Let A be an invertible matrix. show that if λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^-1 PLEASE HELP ME . Thank you.

Homework Statement The question says for the hamiltonian \hat{}H+\hat{}H1 calculate the complete energy eigenvalue spectrum. for the ground state show that the result agrees with the one found by the perturbation theory previously. I'd assume \hat{}H here is just the standard...

So I'm working out on a potentials of the type x^2p, and I have a program that solves and gives the eigenenergies for a potential that I have (x^n in general). I noticed that for a ground state the potential x^4 has the smallest eigenvalue : 0.667981 in units where \hbar=m=\omega=1. I...

I'm taking the math subject GRE in just over a year's time... and I was wondering if there are "ideal" algorithms to have in our tool box to do a computation like this quickly. Obviously the type of matrices in a standardized exam are going to be fairly clean or look dirty but have some less...

We are doing Eigenvalue problems in my Differential Equations class and I just want to make sure I understand some of these concepts. If anyone could look through my current understanding and guide me in the right direction that would be great! So when you have some equation L[y]=\lambda...

Homework Statement For a material the stress is defined by the means of the stress matrix O O = (6 1 -2 1 2 2 -2 2 5) Expressed in MPA It can be derived that the principe stress are: O1= 4-sqrt(13), O2= 5 and O3=4+sqrt(13) I know you can derive the principal...

Homework Statement Solve the differential equation eigenvalue problem: f'' + \lambda f = 0, \quad 0 \leq x \leq 4\pi, \quad \text{where} \quad f^{'}(0) =0, \quad f^{'}(4\pi) = 0, \quad \text{and} f \neq 0. Consider ONLY \quad \lambda \geq 0, \quad and find the values of \quad \lambda...

Assuming that k\geq0, How does one prove that when A has an eigenvaule \lambda that A^{k} has an eigenvalue \lambda^{k}?

I have a PDE test next week and I'm kinda confused. How do you prove that eigenvalues are all positive? I know Rayleigh Quotient shows the eigenvalues are greater than or equal to zero, but can someone explain the next step. Thanks in advance