Eigenvalue Definition and 382 Threads

Homework Statement The Hamiltonian for a two state system is given by H=a(|1><1|-|2><2|+|1><2|+|2><1|) where a is a real number. Find the energy eigenvalue and the corresponding energy eigenstate.Homework Equations The Attempt at a Solution I don't know how to start, I'm looking for a hint...

Hello everybody, I have been trying to solve coupled two eigenvalue (Sturm-Liouville) problems in terms of two (eigen) functions u[x,y] and v[x,y]. I have been using Mathematica trying to solve the coupled equations analytically in their original form, but the Mathematica doesn't seem to...

Homework Statement Given that q is an eigenvalue of a square matrix A with corresponding eigenvector x, show that qk is an eigenvalue of Ak and x is a corresponding eigenvector. Homework Equations N/A The Attempt at a Solution I really haven't been able to get far, but; If x is an...

I'm wondering if anybody could suggest some techniques that might be brought to bear on the following problem: Suppose a finite sequence M_1,M_2,\dots,M_k of 4\times 4 orthogonal reflection matrices is given. I'm interested in determining conditions on these matrices that will guarantee that...

"solution to an eigenvalue problem" ? I am trying to reproduce the results from a paper. The essence of the paper is hidden in just one equation (eq. 11) and some lines of text. For me this is going somewhat too fast. Below are the essential parts of the paper, describing the problem (and...

Homework Statement For the following linear system: \frac{dx}{dt} = -2x \frac{dy}{dt} = -2y Obtain the general solution. Homework Equations The Attempt at a Solution A= -2 0 0 -2 Using the determinant of A-\lambdaI I got a repeated eigenvalue of -2. I am...

Homework Statement There is an Hamiltonian operation which is given by (2 1 1) (1 2 1) = H ; 3-by-3 matrix (1 1 2) And let's have an arbtrary eigenvector (a) (b) = v ; (3x1) matrix (c) Then, from the characteristic equation, the eigenvalues are 1,4. Here eigenvalue 1 is...

Hi, I have this problem on a past exam paper I am having some trouble with: "in the conventional basis of the eigenstates of the Sz operator, the spin state of a spin-1/2 particle is described by the vector: u = \left( \stackrel{cos a}{e^i^b sina} \right) where a and B are constants...

Hi. I was wondering if anyone can give me advice on how to answer the following question. Use Gerschgorin's theorem to show the effect of increasing the size of the matrix in your solution to the eigenvalue problem: y''+lambda*y=0 y(0)=y(1)=0 Thanks Main issue is that I don't...

[URGENT] another Eigenvalue problem Homework Statement [PLAIN]http://img99.imageshack.us/img99/1762/222n.png Homework Equations N/A The Attempt at a Solution I've no clue what's going on for this one. What does that function even do anyway?

[URGENT] Eigenvalue problem Homework Statement [PLAIN]http://img228.imageshack.us/img228/4990/111em.png Homework Equations Sturm-Liouville equation? The Attempt at a Solution I guess I'm just totally lost here. I've no idea how to start. It seems to me that maybe solving for...

Let \lambda be an eigenvalue of A and let \mathbf{x} be an eigenvector belonging to \lambda. Use math induction to show that, for m\geq1, (\lambda)^m is an eigenvalue of A^m and \mathbf{x} is an eigenvector of A^m belonging to (\lambda)^m. A\mathbf{x}=\lambda\mathbf{x} p(1)...

Hi all, Let's say we have a symmetric matrix A with its corresponding diagonal matrix D. If A has only 1 eigenvalue, how do we show that there exists 2 eigenvectors? thanks!

Homework Statement [PLAIN]http://img28.imageshack.us/img28/5227/79425145.jpg The Attempt at a Solution I'm not exactly sure how to go about this problem. How do I start?

Homework Statement If v is an eigenvector of A with corresponding eigenvalue \lambda and c is a scalar, show that v is an eigenvector of A-cI with corresponding eigenvalue \lambda-c. Homework Equations The Attempt at a Solution I started out thinking that I have to figure out how...

does anyone know any free online eigenvalue solver? thanks.

In my literature reviews I found a few things that I can't quite understand. Homework Statement I have the following equation: http://img717.yfrog.com/img717/6416/31771570.jpg I'm told that by using the eigenvalue factorization: http://img89.yfrog.com/img89/760/83769756.jpg , I can...

Homework Statement Hello, I have the following problem: Suppose A is a hermitian matrix and it has eigenvalue \lambda <=0. Show that A is not positive definite i.e there exists vector v such that (v^T)(A)(v bar) <=0 The Attempt at a Solution Let w be an eigenvetor we have the following...

The last matrix at the bottom of the second page is the Eigenvector found using Matlab. I'm trying to find it by hand. I found the Real Eigenvector associated with L=76.2348. But I've tried to find the Eigenvector's for the complex Eigenvalues for a while and can't get the answer given by...

Hi, this is my 1st post here and i was wondering if I could get some help Suppose wehave a 2x2 matrix A with one eigenvalue \lambda, but it is not a scalar matrix. Suppose \vec{v2} is a nonzero vector which is not an eigenvector of A; show that \vec{v1} = (A-\lambda)\vec{v2} is an eigenvector...

Hi all, any help greatly appreciated. Please bear in mind that i have no experience in any Physics and genuinly have no idea how to these questions. Homework Statement An eigenfunction of the operator d^2/dx^2 is ψ = e^2x. Find the corresponding eigenvalue. Homework Equations...

Homework Statement Prove that a square matrix is not invertible if and only if 0 is an eigenvalue of A.Homework Equations The Attempt at a Solution Given: A\vec{x} = \lambda\vec{x} \Rightarrow A\vec{x} - \lambda\vec{x} = \vec{0} \Rightarrow (A - \lambda I)\vec{x} = \vec{0}...

Hi all, the annihilation operator satisfies the equation \hat{a}|n>=\sqrt{n}|n-1> and \hat{a}|0>=0 so the matrix of \hat{a} should be http://www.tuchuan.com/a/2010020418032158925.jpg and zero is the only eigenvalue of this matrix. The coherent state is defined by...

Homework Statement Let A denote a 3x3 matrix with positive real entries. Show that A has a positive real Eigenvalue. Homework Equations This is a problem from a topology course, assigned in the chapter on fundamental groups and the Brouwer fixed point theorem.The Attempt at a Solution I...

This is a revision problem I have come across, I have completed the first few parts of it, but this is the last section and it seems entirely unrelated to the rest of the problem, and I can't get my head around it! Suppose that the 2x2 matrix A has only one eigenvalue λ with eigenvector v...

On the multiplicity of the eigenvalue Dear friends, Might you tell me any hint on the multiplicity of the max eigenvalue, i.e., one, of the following matrix. 1 0 0 0 0 p21 0 p23 0 0 0 p32 0 p34 0...

It was pretty cool to stumble upon Euler's formula as the eigenvalues of the rotation matrix. det(Rot - kI) = (cos t - k)2 + sin2t =k2-2(cos t)k + cos2t + sin2t =k2-2(cos t)k + 1 k = {2cos t +/- \sqrt{4cos^2(t) - 4}}/2 k = cos t +/- \sqrt{cos^2(t) - 1} k = cos t +/- \sqrt{cos^2(t) - cos^2t -...

If a n x n matrix A has an eigenvalue decomposition, so if it has n different eigenvalues, by the way, is it correct that a n x n matrix that doesn't have n different eigenvalues can't be decomposed? Are the more situations in which it can't be decomposed? Why can't I just put the same...

Let A and B be nxn matrices, where B is invertible. Suppose that 4 is an eigenvalue of A, and 5 is an eigenvalue of B. Find ALL true statements. A) 4 is an eigenvalue of A^T B) 4 is an eigenvalue of (B^−1)AB C) 265 is an eigenvalue of (A^4)+A+5I D) 8 is an eigenvalue of A+(A^T) E) 20 is an...

Homework Statement Let A and B be n x n matrices, where B is invertible. Suppose that 2 is an eigenvalue of A, and −2 is an eigenvalue of B. Find ALL true statements below. A. −4 is an eigenvalue of AB B. 16 is an eigenvalue of A^3+A+6I C. 4 is an eigenvalue of A+A(Transpose) D. 2 is an...

dear all how do you find the eigenvalues and eigenvectors of a complex matrix? 0 ; -i ; 0 ; 0 i ; 0 ; -i*sqrt(2) ; 0 0 ; i*sqrt(2) ; 0 ; -i*sqrt(5) 0 ; 0 ...

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