Eigenvalues Definition and 853 Threads

Homework Statement Find the eigenvalues and eigenfunction for the BVP: y'''+\lambda^2y'=0 y(0)=0, y'(0)=0, y'(L)=0 Homework Equations m^3+\lambdam=0, auxiliary equation The Attempt at a Solution 3 cases \lambda=0, \lambda<0, \lambda>0 this first 2 give y=0 always, as the only...

Homework Statement The matrix A = \begin {bmatrix} 0 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 1 & 0 \end{bmatrix} has two real eigenvalues, one of multiplicity 2 and one of multiplicity 1. Find the eigenvalues and a basis of each eigenspace.Homework Equations N/AThe Attempt at a Solution I've done...

Hi there! Let A be a square matrix of order n. It is well known that if we have n distinct eigenvalues then we surely have n distinct eigenvectors. But if there are repeated eigenvalues then the tow possibilities may happen. My question is: How can I know that do the eigenvectors are...

Homework Statement I'm having a problem with a question. I need to find the transition matrix in the form T=UAU^-1 where U=[V1 V2] Homework Equations T=UAU^-1 where U=[V1 V2] The Attempt at a Solution my original transition matrix is [0.9 0.002; 0.1 0.998] from that i calculated...

Homework Statement Given an unknown matrix with eigenvalues 1,2,3, prove that it is invertible? The Attempt at a Solution If the det = 0, then there exists an eigenvalue = 0. Since none of the eigenvalues are 0, then the det ≠ 0 and thus the matrix is invertible. Is this a valid proof?

Homework Statement Find the eigen values of the following mapping and determine if there are invariant lines. (2 -4) (-3 3) is the mapping. Homework Equations det (L-λI)=0 The Attempt at a Solution L-λI= (2-λ -4) (-3 3-λ) det(L-λI)=0=ac-bd=(3-λ)(2-λ)-12 ...

To find the eigenvalues \lambda of a matrix A you solve the equation det |A - \lambda I| = 0 eq(1) but now what if you add e I to the matrix A where e is a constant? Then you have to solve the equation, det |(A + eI) - \lambda_{new} I| = 0 eq(2) which is the same as solving...

Homework Statement Come up with a 2 x 2 matrix with 2 and 1 as the eigenvalues. All the entries must be positive. Then, find a 3 x 3 matrix with 1, 2, 3 as eigenvalues. The Attempt at a Solution I found the characteristic equation for the 2x2 would be λ2 - 3λ + 2 = 0. But then I couldn't get...

Homework Statement Show that y''+\lambda y=0 with the initial conditions y(0)=y(\pi)+y'(\pi)=0 has an infinite sequence of eigenfunctions with distinct eigenvalues. Identify the eigenvalues explicitly.Homework Equations The Attempt at a Solution \lambda \le 0 seems to yield the trivial...

Homework Statement I need to find the general solution of the system [3 5] [-1 -2] Homework Equations so to get the eigenvalues, det(A - λI) The Attempt at a Solution determinant is (3-λ)(-2-λ) + 5 which would be λ2 - λ - 1 so by the quadratic equation the eigenvalues are...

Hello, I have a linear algebra problem that I need help with. Basically, I need to get the eigenvalues and eigenvectors of several (sometimes tens of thousands) very large matrices (6^n x 6^n, where n>= 3, to be specific). Currently, we are just using MATLAB's eig() function to get them. I...

Assuming we have a closed loop system (A-BK), with stable eigenvalues, how would one choose matrices Q and R such that the eigenvalues of (A-BK) are exactly [-1,-2]? LTI System: \dot{x}=\left[ \begin{array}{cc} 0& 1 \\ 0 & 0 \\ \end{array} \right]x+\left[ \begin{array}{c} 0 \\ 1 \\...

Some algorithms of finding the eigenvalues of symmetric matrices first transform the matrix to a tridiagonal matrix which is similar to the original matrix and then find the eigenvalues of the tridiagonal matrix. . Are there special algorithms for a tridiagonal matrix, or do the same algorithms...

Homework Statement Identify the stable, unstable and center eigenspaces for \dot{y} = the 3x3 matrix row 1: 0, -3, 0 row 2: 3, 0, 0 row 3: 0, 0, 1 Homework Equations The Attempt at a Solution This is an example used from the lecture and I understand how to get the...

Homework Statement Apply the eigenvalue method to find a general solution of the given system. x_1' = 5x_1 - 9x_2 x_2' = 2x_1 - x_2 Homework Equations (A-λI)v=0 The Attempt at a Solution x_1' = 5x_1 - 9x_2 x_2' = 2x_1 - x_2 \left[ \begin{array}{cc} 5-λ & -9\\ 2 & -1-λ...

Homework Statement Let λ_n denote the nth eigenvalue for the problem: -Δu = λu in A, u=0 on ∂A (*) which is obtained by minimizing the Rayleigh quotient over all non-zero functions that vanish on ∂A and are orthogonal to the first n-1 eigenfunctions. (i) Show that (*) has no...

Homework Statement (attached)Homework Equations The Attempt at a Solution I really don't know where to start. There is nothing given for me to start with. And the instruction says "Choose" so am I really suppose to really choose or do you guys any idea how to start this? *I know that...

Homework Statement Find the allowed energies for a spin-3/2 particle with the given Hamiltonian: \hat{H}=\frac{\epsilon_0}{\hbar}(\hat{S_x^2}-\hat{S_y^2})-\frac{\epsilon_0}{\hbar}\hat{S_z} The Attempt at a Solution The final matrix I get is: \begin{pmatrix} \frac{3}{2} & 0 &...

Homework Statement Looking for some help with the proof if possible. Vector r = x y z Rotation R = cos(θ) 0 sin(θ) 0 1 0 -sin(θ) 0 cos(θ) r' = Rr It asks me to prove that r'.r' = r.r Second part of the question is about eigenvalues, it asks me to find the three...

Hello everyone, Before I ask my question, be informed that I haven't had any formal course in linear algebra, so please forgive me if the question has a well-known answer. I have two symmetric matrices, A and B. We know the eigenvalues and eigenvectors of A, and B. Now I need to...

Homework Statement Determine the general solution of the system of homogeneous differential equations. The system of homogeneous differential equations is: X'_{1}(t) = 141x_{1}(t) - 44x_{2}(t) X'_{2}(t) = 468x_{1}(t) - 146_{2}(t) What is Eigenvalues of Coefficient Matrix? What is...

Homework Statement This problem will guide you through the steps to obtain a numerical approximation of the eigenvalues, and eigenvectors of A using an example. We will define two sequences of vectors{vk} and {uk} (a) Choose any vector u \in R2 as u0 (b) Once uk has been determined, the...

Find the eigenvalues and corresponding eigenvector of the matrix. A= [-4 4 8 ] [0 0 -10] [0 0 2 ] [1 -1 0] ~ [0 0 1 ] [0 0 0 ] I calculated by A = -\lambdaI So, [1-lamda -1 0 ] [0 -lamda 1] [0 0 -lamda] so, lamda = 0,0, and 1 So I got...

Hey guys, I need to find the equilibrium solution (critical point) for the given system. Also I need to take the homogeneous equation x' = Ax (matrix notation) and find the eigenvalues and eigenvectors. system: x' = -x - 4y - 4 y' = x - y - 6 Can you help? Thanks

Quick question: I have a characteristic polynomial: λ2 + i = 0...how do I solve for the eigenvalues? They're suppose to be + or - (√2/2)(1 - i) How'd they get those?

One more question please... which one of these statements is NOT true (only one can be false): a. a square matrix nXn with n different eigenvalues can become diagonal. b. A matrix that can be diagonal is irreversible. c. Eigenvectors that correspond to different eigenvalues are linearly...

Homework Statement The Attempt at a Solution T(1,0,0) = (3,-1,0) T(0,1,0) = (0,1,0) T(0,0,1) = (-1,2,4) Thus, we have the matrix, \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right| Δ_T (t) = det( \left| \begin{array}{ccc} 3 &0&-1 \\ -1&1&2 \\ 0&0&4 \end{array} \right|...

We are aware that by knowing the eigenvalues and eigenvectors we can evaluate the determinant, say if it is invertible and diagonalize to find powers of matrices. Is there a list of properites of a matrix we can find by eigenvalues and eigenvectors? Are there things that e.values and e.vectors...

Homework Statement The Attempt at a Solution So I observed: T(B) = λB T-1(B) = λ'B Also, T-1(T(B)) = λ'λB = B This implies, λ'λ = 1 And so, there should be a relation λ = \frac{1}{λ'}. Is that right?

I am solving an eigenvalue problem -- Hamiltonian problem in Quantum Mechanics. The matrix is 8x8 with off-diagonal terms, but some are zero. It is well known that the eigenvalues of a Hermitian matrix anti-cross as it nears each other. This is very easy see if the matrix have an independent...

First of all, thanks for all the helpful comments to my previous posts. I'm trying to get a grasp of stress tensors and have been doing some studying. In the literature I've been looking at, it says something about the eigenvalues of stress tensors and the principle stresses. This is...

I've just recently been introduced to charge conjugation while reading the introductory particle physics texts by Griffiths and Perkins, and neither one really seem to explain how you go about finding the values for C. For example, if I wanted to find the value for the \rho^0 meson (which I...

Homework Statement For two endomorphisms ψ and φ on a vector space V over a field K, show that ψφ and φψ have the same eigenvalues. "Hint: consider the cases λ=0 and λ≠0 separately." The Attempt at a Solution I know that similar endomorphisms (φ and ψφ(ψ^-1)) have the same...

Homework Statement dy/dx = y^3-3y^2+2y it's asking for equilibrium points and for the eigenvalues and stability at each point. Homework Equations The Attempt at a Solution I found the equilibrium points by setting dy/dx = 0 as we were taught to do in class and got y = 0, 1, 2...

Homework Statement A) Let A be a symmetric, irreducible, tridiagonal matrix. Show that A cannot have a multiple eigenvalue. B) Let A be an upper Hessenberg matrix with all its subdiagonal elements non-zero. Assume A has a multiple eigenvalue. Show that there can only be one eigenvector...

In my example I have matrix A = (1 2) . . . . . . . . . . . . . . . . . . . . . . (3 2) Finding the eigenvalue through the method I understand & can get the result i.e. k = 4 & -1 I suspect my algebra is the shaky link, here, but to find the eigenline I find a bit more of a...

Hi, I am aware of the implicit QR algorithm, which utilises the 'Francis QR step' to find the eigenvalues of a real, square matrix. But, how would one find the eigenvalues of a complex matrix? Would the 'explicit' version of the QR algorithm be used here, using complex arithmetic? Thanks

Homework Statement I'm working on a problem that involves solving for the eigenvalues and the natural frequencies. I've attached my work as a pdf file and also the MATLAB code used to get the result. The problem that I'm running into is that the frequencies computed from the determinant are...

Homework Statement Potential of a simple harmonic oscillator is \frac{1}{2}m\omega ^{2}(x^{2}+4y^{2}).Find the energy eigenvalues? Homework Equations Seperation of variables,i think. But i got stuck in the midway. The Attempt at a Solution \frac{-\hslash ^{2}}{2m}\left(...

Eigenvalues of A* and A Show that the eigenvalues of A* are conjugates of the eigenvalues of A. I know this is an easy problem, but I've just been spinning my wheels manipulating the equations with the transpose, conjugate, and adjoint properties. \begin{align} A^* = \bar{A}^T\\...

Hello, sorry that I am asking too many questions, I am preparing for an exam... I have a matrix, 0 1 0 0 0 0 0 0 1 and I need to say if it has a diagonal form (I mean, if there are P and D such that D=P^-1*D*P) I found that the eigenvalues are 0 and 1. I also know that if I use 0, I get the...

Homework Statement Calculate the eigenvalues and eigenvectors of the operator, J.n, where n is a unit vector characterized by the polar angles theta and phi, and J is the spin-1 angular momentum operator. Homework Equations Matrix representations for J^2 and J(z) The Attempt at a...

Homework Statement Find wave functions of the states of a particle in a harmonic oscillator potential that are eigenstates of Lz operator with eigenvalues -1 h , 0, 1 h and have smallest possible eigenenergies. Check whether these states are also the eigenstates of L^2 operator. Eventually...

Homework Statement an operator for a system is given by \hat{H}_0 = \frac{\hbar \omega}{2}\left[\left|1\right\rangle\left\langle1\right| - \left|0\right\rangle\left\langle0\right|\right] find the eigenvalues and eigenstates Homework Equations The Attempt at a Solution so i...

Homework Statement Find the eigenvalues and normalized eigenvectors of the rotation matrix cosθ -sinθ sinθ cosθ Homework Equations The Attempt at a Solution c is short for cosθ, s is short for sinθ I tried to solve the characteristic polynomial (c-λ)(c-λ)+s^2=0, and...

Homework Statement Ok I was working with finding Jordan canonical form... Here is the matrix I was working on: | 1 1 1 | |-1 -1 -1 | | 1 1 0 | I am having problem with finding eigenvalues... below is the attempt to solution I was not getting the right answer. So, when I used...

Homework Statement Calculate the Eigenvalues and eigenvectors of H= 1/2 h Ω ( ]0><1[ + ]1><0[ ) Homework Equations I know H]λ> = λ]λ> The Attempt at a Solution I don't know if I am meant to concert my bra's and ket's into matrices, and if so how to represent these as matrices?

I have no trouble calculating eigenvalues but I have a hard time understanding how to use them. I know that you can somehow calculate a bridge's self-frequency with eigenvalues but I don't know how. What I am after is, what do eigenvectors and eigenvectors mean physically or in other ways? I...

According to theory the eigenvalues of a completely disconnected graph (no two nodes are connected) must be all 0. But the normalized Laplacian of such a graph will be an identity matrix whose eigenvalues will be all 1s. Please correct me!

This is related to spectral graph theory. I am getting the following eigenvalues for a 400x400 matrix which is a normalized laplacian matrix of a graph. The graph is not connected. So why am i getting a> a negative eigenvalue. b> why is not second eigenvalue 0? ... I used colt(java) and octave...