Eigenvectors Definition and 452 Threads

Homework Statement For the following operator represented in the orthonormal basis {|1>, |2>} \hat{M} = \begin{pmatrix} 2 & i\sqrt{2} \\ -i\sqrt{2 & 2} \end{pmatrix} find the eigenvalues and eigenvectors and express them as a function of |1> and |2> normalized.The Attempt at a Solution I'm...

Homework Statement I know the eigenvalues, and i have derived the general equation as follows: -x1 + x2 = 0 x1 - x2 = 0 0.x3 = 0 What is the eigenvector? The attempt at a solution I am trying to find the exact procedure to solve this, but there seems to be none. As a general rule...

Homework Statement V is a vector space consisting all functions f:R->R that is differentiable many times (a) Let T:V->V be the transformation T(f)=f' Find the (real) eigenvectors and eigenvalues of T (b) Let T be transformation T(f)=f" Prove that all real number, m is the eigenvalue of...

Homework Statement This is the matrix A, which i need to find the eigenvalues and eigenvectors. 3x3 matrix 5 6 12 0 2 0 -1 -2 -2 The attempt at a solution I have found the eigenvalues to be: 1, 2, 2. So, the final eigenvalues are : 1 and 2. Now, i found the eigenvector for...

I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?

Homework Statement Solve the coupled mass spring problem for two different masses. Similar to the Shankar example: (d/dt)2x1=-2*w1*x1 + w1*x2 (d/dt)2x2=w2*x1 -2*w2*x2 where w1= k/m1 w2 = k/m2 Homework Equations Eigenvalue problem: UX=uX Diagonalization: A=UDUt Exponential...

Homework Statement Let T: M22 - M22 be defined by T(a b = 2c a + c c d) b-2c d (a) Find the eigenvalues T. (b) Find bases for the eigenspaces of T. Homework Equations det(lambdaI - A) = 0 The Attempt at a Solution I just wanted to know how to start here. I am...

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)...

Hey guys (this is not a HW problem, just general discussion about the solution that is not required for the assignment), So I am doing this problem where I had to find the eigenvalues and eigenvectors of the Hamiltonian: H = A*S_{x}^{2} + B*S_{y}^{2} + C*S_{z}^{2}. Easy enough, just...

Hello i have this matrix \in Z mod 7, M = \begin{pmatrix} 0&6\\ 5&0 \end{pmatrix} always modulo 7 in Z. I found characteristic polynomial x^2+5. Eigenvalues are \lambda = 3, \lambda' = 4 Eigenvectors related to \lambda = 3 are the non-zero solution of the system: 4x +6y = 0, 5x+4y = 0 I...

Homework Statement Let B = 1 0 6 -1 Be a square matrix. Find a non-singular matrix P such that P-1BP = D, where D is a diagonal matrix and show that P-1BP = D. Homework Equations det(lambdaI - A) = 0 The Attempt at a Solution Ok, this might look like a simple...

Hello all. This is my first post here. Hope someone can help. Thank you guys in advance. Here is the question: I have a n-by-n matrix A, whose eigenvalues are all real, distinct. And the matrix is positive semi-definite. It has linearly independent eigenvectors V_1...V_n. Now I have known...

Homework Statement Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4] Homework Equations det(A - lambda I) = 0 The Attempt at a Solution After expanding using the formula I have the equation (2 - \lambda)(4 - \lambda) - 8 Which gives \lambda = 0, 6 (Should I...

Homework Statement So the 3x3 matrix involved is [3 -1 -1:-4 6 4:-1 1 1], The eigenvalues are L=6 and L=2. Homework Equations (A-LI)e=0 The Attempt at a Solution I stuck the eigenvalues into the matrix and got (-1 1 1)(not sure if its right) for L=2 but when I use L=6 in I can't...

Homework Statement This question is from Principles of Quantum Mechanics by R. Shankar. Given the operator (matrix) \Omega with eigenvalues e^{i\theta} and e^{-i\theta} , I am told to find the corresponding eigenvectors.Homework Equations \Omega = \left[ \begin{array}{cc} \cos{\theta}...

Homework Statement Im looking at finding the eigenvectors of a matrix but also a basis for the eigenspace A = [ 6 16 ] [ -1 -4 ] lambda = 4 lambda = -2 Homework Equations (A - lambda I ) v = 0 The Attempt at a Solution So with the above equation I get: for...

Homework Statement Being T\in L(\mathbb{R}^n) a linear operador defined by T(x_1, ... ,x_n )=(x_1+...+x_n,...,x_1+...+x_n ), find all eigenvalues and eigenvectors of T. Homework Equations det(T-\lambda I)=0, Ax=\lambda x The Attempt at a Solution By checking n=1,2,3,4 I guess the...

Homework Statement I'm trying to use the transfer matrix method in statistical mechanics but I'm struggling with the algebra so I'd like to know if there is a simpler way to find the eigenvalues and eigenvectors of a matrix. For example, studying the lattice gas model produces the transfer...

Given the matrix: \left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) I'll find eigenspace for the eigenvalue of t=0, So I have to solve: \left( \begin{array}{cc} 1 & 2 \\ 2 & 4 \end{array} \right) {(x,y)}^t=0 Then I do R_2->R_2-2R_1 and get x+2y=0 => x=-2y => get the eigenvector...

Hi all, I'm trying to find the eigenvectors of a symbolic 3x3 rotation matrix in MATLAB, it appears to work for some inputs but not all, for example: A =[ cos(q), -sin(q), 0] [ sin(q), cos(q), 0] [ 0, 0, 1] [V,lambda]=eig(A) V =[ 0, -i, i] [ 0, 1, 1]...

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...

If v is in Rn and is an eigenvector of matrix A, and P is an invertible matrix, how would you go about finding an eigenvector w of PAP-1? I'm thinking you have to use a fact about similarity?

Homework Statement Find the general solution for the following systems of equations ( 2 0 ) ( 0 2 ) Homework Equations (A-Ix) c1e^at[]+c2e^bt[] The Attempt at a Solution when attempting to find the eigenvalues I come up with (2) so plugging back into get the vectors you come up...

Dear Friends & Colleagues, I have a couple of nagging issues with mathematics I was hoping anyone of you would kindly be able to help resolve. Given a few (let’s say 3) square matrices, A_1, A_2, A_3, and the corresponding principal eigenvector (eigenvector corresponding to the largest...

Homework Statement find the general solution to x'=Ax; where A is a 3x3matrix: A=[0 1 1; 1 0 1; 1 1 0] Homework Equations det(A-lambda*I)=0 The Attempt at a Solution i found the eigenvalues to be 2, -1, -1. for lambda=2 i found the corresponding eigenvector to be a 3x1 martrix...

Homework Statement Let A be a 2x2 matrix with complex eigenvalue (lambda)=a-bi, (b does not equal 0) and associated eigenvector v-> C^2. Show that A(Rev)=aRev +bImv and A(ImV)=-bRev +aImv. Where Re is real part and I am is imaginary part. Homework Equations The Attempt at a...

Let L : V>>>V be an invertible linear operator and let lambda be an eigenvalue of L with associated eigenvector x. a) Show that 1/lambda is an eigenvalue of L^-1 with associated eigenvector x. For this question, the things I know are that L is onto and one to one. Therefore, how to prove this...

Homework Statement Let \[ A = \begin{array}{ccc} 3 & 5 & 6 \\ 0 & 2 & h \\ 0 & 0 & 2 \\ \end{array} \]. What should be the value of h so that there are two linearly independent eigenvectors of A corresponding to eigenvalue 2? Homework Equations (A-\lambda I) \textbf{x} =...

What interpretation could the eigenvectors in a graph have? By graph I mean an adjaceny matrix not counting self-loops. If you can draw any physical meanings or point to any examples, that'd be even better. Thank you!

Homework Statement When generating a matrix from eigenvectors, does it matter in which order the columns are placed?

Homework Statement a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = Hx+Hy+Hz, where Hx=1/2*(px2+x2), Hy=1/2*(py2+y2), Hz=1/2*(pz2+z2) Check that the standard angular momentum operators Lx, is a constant of motion. b.) By knowing that the...

Question: find the eigenvectors of {{2,-1,-1},{-1,2,-1},{-1,-1,2}}. Equations: none Attempted answer: I have the characteristic equation as x^3-6x^2+9 which gives eigenvalues as 0 3 and 3. This is correct says wolfram alpha for x=0 {{2,-1,-1},{-1,2,-1},{-1,-1,2}}•{x,y,z}=0 I get 3...

I am trying to get an eigenvector for the following matrix, I am up to the final step. 4 1 0 0 I got it to be -1 4 is this the same as 1 -4 sorry I am pretty new to linear algebra.

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...

I essentially know how to find eigenvalues and thus eigenvectors, though when solving a problem about diagonalization I do not know how to order them (as in, I can find all the eigenvectors but do not know which order to place them into find my X that diagonalizes my A) In the examples of my...

I am working on problems involving the eigenvalue method for homogenous systems in my differential equations class. However, I have not had linear algebra yet so I am not understanding some of this. I've made it to the step where I have three lambda values and plugging them back into the...

Homework Statement Find the eigenvalues and a corresponding system of eigenvectors of the operator Af(x) := Integration from 0 to 1 K(x; y)f(y) dy where K(x; y) = cos (2pi(x - y))

if we say that an eigenvector u is a vector whose direction is unchanged by the matrix transformation A: Au=λu=λIu rearranging: Au - λIu = 0 u(A-λI)=0 we know the u=0 is obviously a solution. and i also appreciate that if A-λI=0 then A=λI which is just a scalar multiplied by the identity so is...

Homework Statement Find the eigenvalues and an orthonormal set of eigenvectors for this matrix: |1 1+i 0| |1-i 1 0| |0 0 2| Homework Equations Find the determinant of A - xI, where A is the matrix, I is the identity matrix, and x denotes eigenvalues Set the...

Homework Statement Okay, we've got a 3x3 matrix. All the entries are equal to 1. Call the matrix A. Basically It looks like this 1 1 1 1 1 1 1 1 1 find all the eigenvalues and eigenvectors. The Attempt at a Solution Right, I got the eigenvalues being 0 and 3, the same as my...

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...

Is there such a thing as a square matrix with no eigenvectors? I'm thinking not ... since even if you have: \left[\begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array}\right] you could just as well say that the eigenvalue(s) are 0 (w/ algebraic multiplicity 2) and the eigenvectors are: u_1 =...

Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen2.jpg The second term should be De^( - √(5)t), I made a mistake when writing out the question. The Attempt at a Solution I worked it out to be A = 0, B = -1/5, C = 3/10, D = -3/10 answer is 3.09 Is that correct...

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) =...

I see that a generalized eigenvector can be represented as such: (A - λI)xk+1 = xk, where A is a square matrix, x is an eigenvector, λ is the eigenvalue I is the identity matrix. This might be used, for example, if we have duplicate eigenvalues, and can only derive one eigenvector from...

Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?

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 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...

I am evaluating the following 2 x 2 matrix: |2 0 | |0 3 | with eigenvalues 2 and 3. If I use 2 and calculate the eigenvector: R - λI = |2-λ 0 | |0 3-λ | R - λI = |0 0 | |0 1 | |0 0 ||a| = |0 1 ||b| |0| |0| a = 0 and b = 1 So...