Eigenvalues Definition and 820 Threads

Homework Statement http://img820.imageshack.us/img820/4874/cah.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Did it already, 3 is the eigenvalue b) This is just finding the nullspace and the basis of the nullspace are my eigenvectors right? c) ignore...

given that 2 matrices have the same eigenvalues is it necessary that they be similar? If not, what is the connection between those 2?

Suppose I get the eigenvalues of A, which are \lambda_{1},\lambda_{2},\dots \lambda_{n}. Also, given any polynomial f(x), I get the eigenvalues of f(A). I'm trying to show that the eigenvalues of f(A) are f(\lambda_{1}),f(\lambda_{2}),\dots f(\lambda_{n}). Is this possible? How would I go about...

I am trying to relate eigenvalues with singular values. In particular, I'm trying to show that for any eigenvalue of A, it is within range of the singular values of A. In other words, smallestSingularValue(A) <= |anyEigenValue(A)| <= largestSingularValue(A). I've tried using Schur...

1.\frac{dx}{dt}= \stackrel{9 -12}{2 -1} x(0)=\stackrel{-13}{-5} So I seem to be having issues with this problem There are 2 eigenvalues that I obtained from setting Det[A-rI]=0 That gave me r^{2}-8r+15=0 solving for r and finding the roots i got (r-3)*(r-5)=0 so the...

Homework Statement using X''(x)+ lambda*X(x)=0 find the eigenvalues and eigenfunctions accordingly. Use the case lambda=0, lambda=-k2, lambda=k2 where k>0 Homework Equations X(0)=0, X'(1)+X(1)=0 The Attempt at a Solution I know that for lambda=0 X(x)=C1x+C2 which applying the...

Hi all, I am trying to find numerically the eigenvalues of a nonlinear schroedinger equation in a similar way as the Self Consistent Field method for Hatree-Fock problems. Does anybody know in the SCF calculation how to improve the convergency? Is there any trick other than simply inserting...

Homework Statement y'' + (lambda)y = 0, y'(0) = 0, y(1) = 0 We are told that all eigenvalues are nonnegative. Even with looking at the solution manual, I am unsure how to start setting these up. I've been starting by doing the following: y(x) = A cos cx + B sin dx y'(x) = -Ac sin(cx) + Bd...

Homework Statement See figure attached Homework Equations The Attempt at a Solution \lambda > 1, y^{''} + 2y^{'} + \alpha^{2}y = 0, \quad \alpha > 0 Into auxillary equation, m^{2} + 2m + \alpha^{2} = 0 I'm stuck as to how to solve this auxillary equation. Any...

Homework Statement Let F be a finite field of characteristic p. As such, it is a finite dimensional vector space over Z_p. (a) Prove that the Frobenius morphism T : F -> F, T(a) = a^p is a linear map over Z_p. (b) Prove that the geometric multiplicity of 1 as an eigenvalue of T is 1. (c) Let F...

So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand...

I need to know if there is any relationship between the positive definite matrices and its eigenvalues Also i would appreciate it if some one would also include the relationship between the negative definite matrices and their eigenvalues Also can some also menthow the Gaussian...

Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...

The usual eigenvalues of a LTV system does not say much about the stability but my intuition tells me there should be some kind of extension that applies to LTV systems as well. Like including some kind of inner derivative of the eigenvalues or something, I don't know... I guess in some way...

Homework Statement Matrix A has eigenvalues \lambda1= 2 with corresponding eigenvector v1= (1, 3) and \lambda2= 1 with corresponding eigenvector v2= (2, 7), find A. Homework Equations Definition of eigenvector: Avn=\lambdanvn The Attempt at a Solution I tried this by making...

Homework Statement Let V be the linear space of all real polynomials p(x) of degree < n. If p \in V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? Homework Equations Not sure...

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

Homework Statement Consider the Hamiltonian \hat{}H = \hat{}p2/2m + (1/2)mω2\hat{}x2 + F\hat{}x where F is a constant. Find the possible eigenvalues for H. Can you give a physical interpretation for this Hamiltonian? Homework Equations The Attempt at a Solution I don't think...

Homework Statement Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V: a) G(f) = xD(f), where f is an element of V and D is the differentiation map...

Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen.jpg The Attempt at a Solution It is a bit too long to type it all out, but I was wondering whether I am correct: I got, A = 7/2 , B = 0 , C = -1/8 , D = 1/8 And from this I worked out, x2(1) =...

Homework Statement dx/dt= -4x -y dy/dt= x-2y x(0)=4 y(0)=1 x(t)=? y(t)=? Homework Equations The Attempt at a Solution 1) find eigenvalues (x+4)(X+2)+1 X=-3,-3 2)eigenvectors: (-3-A)(x,y)=(0,0) eignvector=(-1,1) 3)using the P from this page...

From my Linear Algebra course I learned tha and eigenvalue w is an eigenvalue if it is a sollution to the system: Ax=wx, where A= square matrix, w= eigenvalue, x= eigenvector. We solved the system by setting det(A-I*w)=0, I=identity matrix Now in an advanced course I have come upon the...

Find the eigenvalues and corresponding eigenvectors of the following matrix. 1,1 1,1 Here is my attempt to find eigenvalues: 1-lambda 1 1 1-lambda Giving me: (Lambda)^2 -2(lambda) = 0 lambda = 0 lambda = 2 Is this correct??

Homework Statement Show that the eigenvalues of a hermitian operator are real. Show the expectation value of the hamiltonian is real. Homework Equations The Attempt at a Solution How do i approach this question? I can show that the operator is hermitian by showing that Tmn =...

Homework Statement Let A \in \mathbb{C}^{n \times n} and set \rho = \max_{1 \le i \le n}|\lambda_i|, where \lambda_i \, (i = 1, 2, \dots, n) are the eigenvalues of A. Show that for any \varepsilon > 0 there exists a nonsingular X \in \mathbb{C}^{n \times n} such that \|X^{-1}AX\|_2 \le...

Homework Statement Hi there. I must give the eigenvalues and the eigenvectors for the matrix transformation of the orthogonal projection over the plane XY on R^3 So, at first I thought it should be the eigenvalue 1, and the eigenvectors (1,0,0) and (0,1,0), because they don't change. But I...

Homework Statement I am part way done with this problem... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now. W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it. a)...

Find a matrix that has eigenvalues 0,18,-18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1). ... I know the diagonlize rule, and the the rule to find a a power of A A= PDP^-1 D=P^-1AP ... but i am lost as to how to contine... help please?

Homework Statement Find the eigen values, eigenspaces of the following matrix and also determine a basis for each eigen space for A = [1, 2; 3, 4]Homework Equations \det(\mathbf{A} - \lambda\mathbf{I}) = 0 The Attempt at a Solution OK, so I found the eigenvalues and eigenspaces just fine...

Homework Statement If two 3 x 3 matrices A and B have the eigenvalues 1, 2, and 3, then A must be similar to B. True or False and why. Homework Equations A is similar to B iff B = S^-1AS The Attempt at a Solution I know that if A and B are similar then they have the same eigenvalues but the...

Homework Statement Find all non-zero eignvalues and eigenvectors for the following integral operator Kx := \int^{\ell}_0 (t-s)x(s) ds in C[0,\ell] Homework Equations \lambda x= Kx The Attempt at a Solution \int^{\ell}_0 (t-s)x(s) ds = \lambda * x(t)...

Hey folks, I'm having an issue using a routine from the netlib that is supposed to calculate eigenvalues and eigenvectors. The canned routine can be found here: http://www.netlib.org/seispack/rgg.f I want to find the eigenvalues of a matrix (a more complex hamiltonian), so for my simple...

Homework Statement Given the matrix A = [1 0 0 -2 1 3 1 1 -1] Find an invertable matrix X and a diagonal matrix D such that A = XDX^-1 Homework Equations A = XDX^-1The Attempt at a Solution I've found that the eigenvalues are -2, 2...

This is probably falls within a problem of Mathematica as opposed to a question on here but I have a question about the following: Given some cylinder with the shape operator matrix: {{0,0},{0,-1/r}} We get eigenvalues 0 and -1/r and thus eigenvectors {0, -1/r} and {1/r, 0} by my...

Homework Statement solve the system: dx/dt = [1 -4] x _______[4 -7] with x(0) = [3] __________[2] Homework Equations The Attempt at a Solution I got both eigenvalues of the matrix are -3 and so both eigenvectors are [1]...

We have two nxn matrices with non-negative elements, A and B. We know the eigenvalues and eigenvectors of A and B. Can we use this information to say anything about the eigenvalues or eigenvectors of C=A*B? The largest eigenvalue of C and the associated eigenvector are of particular interest...

1. a) The action of the parity operator, \Pi(hat), is defined as follows: \Pi(hat) f(x) = f(-x) i) Show that the set of all even functions, {en(x)}, are degenerate eigenfunctions of the parity operator. What is their degenerate eigenvalue? The same is true for the set of all odd functions...

If I have a matrix for which all eigenvalues are zero, what can be said about its properties? If I multiply two such matrices, will the product also have all zero eigenvalues? Thanks

We have vectors x,y of size n and a matrix A of size nxn. Is it true that the matrix xyTA has at most one non zero eigenvalue? Why is it so?

Homework Statement How do we prove that commuting matrices have common eigenvalues? Homework Equations The Attempt at a Solution

First, I appologise if this is in the wrong place, while the book is QM, the question is pure maths. Also I'm not sure if this techically counts as homework as I am self studying. Finally, sorry for the poor formatting, I'm not that good with LaTeX Homework Statement Given the matrix...

Homework Statement Prove or disprove the title of this thread. Homework Equations AX=(lamda)X The Attempt at a Solution I don't know where to start

http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165642.jpg?t=1287612122 http://i111.photobucket.com/albums/n149/camarolt4z28/2010-10-20165727.jpg?t=1287612136 Thanks.

Hi everyone Consider a 2x2 partitioned matrix as follow: A = [ B1 B2 ; B3 B4 ] I'm sure that all eigenvalues of A are on the unit circle (i.e., abs (all eig) = 1 ). but, I don't know how to prove it. Is there any theorem?

Homework Statement I am wondering if I can make the sweeping generalization that the eigenvalues of the zero ket are zero. I further generalize that the zero ket is not of interest, as far as physical observables occur. Homework Equations the eight axioms of vector spaces...

Lets say I have a 3x3 matrix 'A' and one known eigenvalue 'z' and one known eigenvector 'x', but they don't "belong" to each other, as in Ax =/= zx Is there a way of finding the other eigenvalues and eigenspaces of A using only this piece of information? Thanks.

Hi guys, Can someone please explain how you find the eigenvalues of this type? u''+\lambda u =0 or point me to some decent literature? regards Brendan

hi, not strictly homework as my course doesn't get going again for a couple of weeks yet, but suppose I have a system with quantum number l=1 in the angular momentum state u = \frac{1}{\sqrt{2}} \left(\begin{array}{cc}1\\1\\0\end{array}\right) and I measure Lz, the angular momentum component...

Dear all, in basic QM books the position and momentum operators (continuous eigenvectors) are introduce by means of the dirac delta and some analogies are made with the infinite dimensional, but discrete case in order to provide some intuition for the integral formulas presented. My knowledge...

Homework Statement The energy eigenvalues of a particles of mass, m, confined to a 3-d cube of side a are: E_{nx,ny,nz}=\frac{a(n^{2}_{x}+n^{2}_{y}+n^{2}_{z})}{b}+ Vo where: a= planks constant^2(pi)^2 b=2m^2 nx,ny,nz = any positive integers. What are the ground-state kinetic and potential...