Eigenvalue Definition and 402 Threads

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of **A**. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f)...

Homework Statement A linear transformation with Matrix A = ## \begin{pmatrix} 5&4&2\\ 4&5&2\\ 2&2&2 \end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1. Homework Equations 3. The Attempt at a Solution [/B] I know from the...

Please have a look at the attached images.I am attempting a proof for the statement : The algebraic multiplicity of an eigen value λ is equal to dim null [T - λ I] dim V. Please advise me on how to move ahead. Apparently, I am at the final inference required for a proof but unable to move...

Homework Statement find all eigen-values and eigen-functions for the initial boundary value problem: $$x^2y''+xy'-\lambda y=0$$ Boundary Conditions: $$y(1)=y(e)=0$$ Homework EquationsThe Attempt at a Solution i just wanted to know if my substitution in the Auxiliary equation is...

Hi, i am trying to find the natural period of a vertical cantilever beam which is fixed at bottom and free at other end., i worked out the global M & K matrices and i have the eqn in the form [M]-w^2[K] = 0, the M & K are not diagonal matrices, but square symetric matrices of rank 6. i...

Consider the Hamiltonian ##H = - \frac{d^2}{dx^2}+gx^{2N}##. Scaling out the coupling constant ##g##, the eigenvalues scale as ##\lambda \propto g^{\frac{2}{N+2}}##. So, we can drop the g dependence and just consider the numerical value of the eigenvalues and the associated spectral functions...

Homework Statement Homework Equations The Attempt at a Solution I did Fourier transform directly to the eigenvalue equation and got Psi(p)=a*Psi(0)/(p^2/2m-E) But the rest, I don't even know where to start. Any opinion guys?

Hello, I was wondering if H_{ii} (that is the ith diagonal element of a random matrix) has the same distribution with its corresponding eigenvalue, say \lambda_{i}. Thanks

Hi, I'm wondering what eigenvalue problem solver you are using? I'm looking for an one which could solve a very large eigenvalue problem, the matrices being ~ 100,000*100,000. Do you have any advices? Thanks.

Homework Statement Suppose: Ĥ = - (ħ2/(2m))(delta)2 - A/r where r = (x2+y2+z2) (delta)2 = ∂2/∂x2 + ∂2/∂y2 + ∂2/∂z2 A = a constant Then, show that a function of the form, f(r) = Ce-r/a with a, C as constants, is an EIGENFUNCTION of Ĥ provided that the constant a is chosen correctly. Find the...

Hi, I have a problem with the calculation of the eigenvalue of a matrix. That matrix is an N x N matrix which can be written as: ##M^{ab} = A\delta^{ab} + B \phi^a \phi^b## where ##\delta^{ab}## is the identity matrix and the ##\phi## is a column vector. The paper I'm studying says that...

Homework Statement In the calculation of the Zeeman Effect, the most important calculation is \langle L_z + 2S_z \rangle = \langle J_z + S_z\rangle Suppose we want to find the Zeeman Effect for ##(2p)^2##, meaning ##l=1##. In Sakurai's book, My question is, what is ##m##? They say that...

Let's say my eigenvalue λ=-1 and we assume eigenvector of zero are non-eigenvector. An eigenspace is mathematically represented as Eλ = N(λ.In-A) which essentially states, in natural language, the eigenspace is the nullspace of a matrix. N(λ.In-A) is a matrix. Would it then be valid to say...

I have a problem I need to solve. I can't find anything in my book that tells me how to do it. It might be worded differently in the book, but I'm not 100% sure how to solve this. Homework Statement Give a description of the eigenvectors corresponding to each eigenvalue. The Attempt at a...

Hi all, Generally boundary condition (Dirichlet and Neumann) are applied on the Load Vector, in FEM formulation. The equation i solved, is Generalized eigenvalue equation for Scalar Helmholtz equation in homogeneous wave guide with perfectly conducting wall ( Kψ =λMψ ), and found, doesn't...

In order to apply perturbation theory to the ψ_{200} and ψ_{210} states, we have to solve the matrix eigenvalue equation. Ux=λx where U is the matrix of the matrix elements of H_{1}= eEz between these states. Please see the matrix in attachment 1. where <2,0,0|z|2,1,0>=<2,1,0|z|2,0,0>=3a_{o}...

Hey! :o I have the following exercise and I need some help.. $"\text{The eigenvalue problem } Ly=(py')'+qy=λy, a \leq x \leq b \text{ is of the form Sturm-Liouville if it satisfies the boundary conditions } p(a)W(u(a),v^*(a))=p(b)W(u(b),v^*(b)). \text{ Show that the boundary conditions of the...

Hello guys, suppose we have an eigenvalue problem \left\{ \begin{array}{ll} u'' + λu = 0, \quad x \in (0,\pi) \\ u(0)=0 \quad \\ \left( \int_0^\pi \! {(u^+)}^2 \, \mathrm{d}x \right)^{\frac{1}{2}} = \left( \int_0^\pi \! {(u^-)}^4 \, \mathrm{d}x \right)^\frac...

Could you explain me: what the difference is between singular value decomposition and eigenvalue problem, when square matrices are involved. Thanks

Homework Statement Let there be a 4X4 Matrix A with dim(im(A), or rank = 1 , and trace=10. What are the Eigenvalues of A? Are there any multiplicities? The Attempt at a Solution While I understand that the trace of a matrix that's 4X4 = the sum of the diagonal elements, I'm confused...

Hi everyone, I have a square matrix J \in \mathbb{C}^{2n \times 2n} where, $J=\left(\begin{array}{cc}A&B\\\bar{B}&\bar{A}\end{array}\right)$ A \in \mathbb{C}^{n \times n} and its conjugate \bar{A} are diagonal.Assume the submatrices A,B \in \mathbb{C}^{n \times n} are constructed in a way...

$L: V\to V$ a diagonalizable linear operator on finite-dim vector space. show that $V = C_x$ iff there are no multiple eigenvalues ------- here $C_x = \operatorname{span} \{x, L(x), L^2(x), \cdots\}$ basically it is a cyclic subspace generated by x that belongs to V. edit: solved

Dear all, I have a problem about the eigenvalue of the system and the eigenvalue of the part of the system. For example,in the theory of the APW method,the space of the primitive cell is divided into muffin-tin (MT) spheres and the interstitial region (IR). In order to gain the...

Homework Statement I'm running through practice papers for my 3rd year physics exam on atomic and nuclear physics: This is the operator we found in the previous part of the question L = -i*(hbar)*d/dθ Next, we need to find the eigenvalues and normalised wavefunctions of L The...

Question one: in regards to two segments underlined in blue. If (a,x) is an eigenvalue and vector of A, that means Ax = ax, where a is a real number. My question is, is Amx = amx, where m in an integer greater than 1? Question 2: in regards to two segments underlined in red. I...

I just have a question about the problem for when the eigenvalue = 0 Homework Statement for y_{xx}=-\lambda y with BC y(0)=0 , y'(0)=y'(1) Homework Equations The Attempt at a Solution y for lamda = 0 is ax+b so from BC: y(0)=b=0 and a=a What is the conclusion to...

Homework Statement The probelm is to show, that a rotation matrix R, in a odd-dimensional vector space, leaves unchanged the vectors of at least an one-dimensional subspace. Homework Equations This reduces to proving that 1 is an eigenvalue of Rnxn if n is odd. I know that a rotational...

Are eigenvalue problems and boundary value problems (ODEs) the same thing? What are the differences, if any? It seems to me that every boundary value problem is an eigenvalue problem... Is this not the case?

Homework Statement Prove true or false. If A^2+A=0 then λ=1 may not be an eigenvalue. Homework Equations To find the eigenvalues of A I find the solutions to det(λ-A). The definition of an eigenvalue from my understanding, AX = λX. A(A+I) = 0 The Attempt at a Solution...

My friends and I have been struggling with the following problem, and don't understand how to do it. We have gotten several different answers, but none of them make sense. Can you help us? **Problem statement:** Let $V$ be the vector space of real-coefficient polynomials of degree at most $3$...

I have the following system of partial differential algebraic equations: [ tex ] \frac{1}{H_p}\frac{\partial H_p}{\partial t} = - \frac{\partial W_p}{\partial x} - \frac{f1(H_p,c_p,W_p)}{H_p}, [ \tex ] [tex] \frac{1}{H_p}\frac{\partial}{\partial t}(H_p c_p} = - \frac{\partial}{\partial...

Given \[ f(x) = \lambda\int_0^1xy^2f(y)dy \] At order \(\lambda^2\) and \(\lambda^3\), we have repeated zeros so \[ D(\lambda) = 1 - \frac{\lambda}{4}. \] Then we have \[ \mathcal{D}(x, y;\lambda) = xy^2 \] so \[ f(x) = \frac{\lambda}{D(\lambda)}\int_0^1\mathcal{D}(x, y;\lambda)dy. \] How do I...

Hello MHB, I got one question. If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)? Regards, |\pi\rangle

Given \[ f(x) = \lambda\int_0^1xy^2f(y)dy \] I am trying to determine the eigenvalues and eigenfunction. I know that the \(\frac{1}{\lambda}\) are the eigenvalues. We can write \(f(x) = xA\) and \(A = \lambda\int_0^1y^2f(y)dy\). \[ A\Bigg(1 - \lambda\int_0^1y^3dy\Bigg) = 0\quad (*) \] So is...

I've found the characteristic equation of the system I'm trying to solve: $$ω^{4}m_{1}m_{2}-k(m_{1}+2m_{2})ω^{2}+k^{2}=0$$ I now need to find the eigenfrequencies, i.e. the two positive roots of this equation, and then find the corresponding eigenvectors. I've been OK with other examples...

Homework Statement The angular part of the Schrodinger equation for a positron in the field of an electric dipole moment {\bf d}=d{\bf \hat{k}} is, in spherical polar coordinates (r,\vartheta,\varphi), \frac{1}{\sin\vartheta}\frac{\partial}{\partial\vartheta} \left( \sin\vartheta\frac{\partial...

Homework Statement The Hamiltonian for a particle in a harmonic potential is given by \hat{H}=\frac{\hat{p}^2}{2m}+\frac{Kx^2}{2}, where K is the spring constant. Start with the trial wave function \psi(x)=exp(\frac{-x^2}{2a^2}) and solve the energy eigenvalue equation...

Hi everyone, I have this linear map A:R^3 \rightarrow R^3 I have that A(v)=v-2(v\dot ô)ô); v,ô\in R^3 ;||ô||=1 I know that A(A(v))=v this telling me that A is it's own inverse. From there, how can I find the eigenvalue of A? Thanks

A=a.a', where a is an N by 1 vector,a'a=5,and T is transpose. a)Give the largest eigenvalue of A. b)what is the corresponding eigenvector? Please help me to solve the problem.

there is potential V(x). If at some point x=a wavefunction have some energy eigenvalue, then Is it guaranteed that It has same energy throughout whole region? Where can I find explanation about this?

Hi all, I want to ask a question about the eigenvalue problem (EVP) and the initial value problem (IVP). Let's say we are solving this linear equation \frac{\partial u}{\partial t}=\mathcal{L}u, the operator L is dependent on some parameters like Reynolds number. I first check the...

If two matrices similar to one another are diagonalizable, then certainly this is the case, since the algebraic multiplicity of any eigenvalue they share must be equal (since they are similar), and since they are diagonalizable, those algebraic multiplicities must equal the geometric...

I have two real symmetric matrices A and B with the following additional properties. I would like to know how the eigenvalues of the product AB, is related to those of A and B? In particular what is \mathrm{trace}(AB)? A contains only 0s on its diagonal. Off diagonal terms are either 0 or...

An eigenvector is defined as a non-zero vector 'v' such that A.v = λ.v I don't understand the motive behind this. We are trying to find a vector that when multiplied by a given square matrix preserves the direction of the vector. Shouldn't the motive be the opposite i.e. finding the matrix...

Hello, I am currently teaching myself quantum mechanics using MIT's OCW and am suck on the following problem from the second problem set of the 2005 7.43 class. Homework Statement Consider an operator O that depends on a parameter λ and consider the λ-dependent eigenvalue equation...

Odd Form Of Eigenvalue -- Coupled Masses This isn't strictly homework, since it's something I'm trying to self-teach, but it seems to fit best here. Homework Statement It's an example of applying eigenvalue methods to solve (classical) mechanical systems in an introductory text to QM...

suppose function f is define on the interval [0,1] it satisfies the eigenvalue equation f'' + E f=0, and it satisfies the boundary conditions f'(0)+ f(0)=0, f(1)=0. How to solve this eigenvalue problem numerically? the mixed boundary condition at x=0 really makes it difficult

Hello, I have a feeling that the solution to this question is going to be incredibly obvious, so my apologies if this turns out to be really dumb. How do I solve the following eigenvalue problem: \begin{bmatrix} \partial_x^2 + \mu + u(x) & u(x)^2 \\ \bar{u(x)}^2 & \partial_x^2 + \mu +...

Homework Statement Find eigenvalue for matrix B= {[3,4,12],[4,-12,3],[12,3,-4]} Homework Equations The Attempt at a Solution I set up the charactersitic polynomial and got the equation: Pa(x) = (x-3)(x+12)(x+4) = x3 + 132 - 144 + 144 = x3 + 132 So I have 3 eigenvalues: 0...

Homework Statement x' = \begin{pmatrix}0&1&3\\2&-1&2\\-1&0&-2\end{pmatrix}*x The Attempt at a Solution I've found the repeated eigenvalues to be λ_{1,2,3}=-1 I've also found the first (and only non zero eigenvector) to be \begin{pmatrix}1&2&-1\end{pmatrix}, but I'm not entirely...