Eigenvector Definition and 144 Threads

Homework Statement Let B = S^-1 * A * S and x be an eigenvector of B belonging to an eigenvalue \lambda. Show S*x is an eigenvector of A belonging to \lambda. Homework Equations The Attempt at a Solution The only place I can think of to start, is that B*x = \lambda*x. However...

I'm a physics major. As such, I have come across several situations in my studies that require the use of eigenvectors and eigenvalues. Whenever I have to use this method, I've been told to. I do not have a complete understanding of eigenvectors and values and am wondering how you would spot a...

Homework Statement The linear operator T on R2 has the matrix \begin{bmatrix}4&-5\\-4&-3 \end{bmatrix} relative to the basis {(1,2), (0,1)} Find the eigenvalues of T, and obtain an eigenvector corresponding to each eigenvalue...

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...

Homework Statement Let A be a matrix corresponding to reflection in 2 dimensions across the line generated by a vector v . Check all true statements: A. lambda =1 is an eigenvalue for A B. Any vector w perpendicular to v is an eigenvector for A corresponding to the eigenvalue lambda =1. C...

Hi! This might be a silly question, but I can't seem to figure it out and have not found any remarks on it in the literature. When diagonalizing an NxN matrix A, we solve the characteristic equation: Det(A - mI) = 0 which gives us the N eigenvalues m. Then, to find the eigenvectors v...

Homework Statement Find the eigenvalues and eigenvectors of \left( \begin{array}{ccc} 2 & 0 & 0 \\ 0 & 3 & 4 \\ 1 & 1 & 0 \end{array} \right) Homework Equations p(\lambda) = det(A - \lambda I) = 0 The Attempt at a Solution A - \lambda I = \left( \begin{array}{ccc} 2-\lambda & 0 & 0 \\ 0...

Homework Statement On page 5 of http://arcsecond.wordpress.com/2009/08/01/the-cauchy-schwarz-inequality-and-heisenbergs-uncertainty-principle/ the author states (w/o proof) that if \psi is an eigenvector (say with eigenvalue \lambda) of an Hermitian operator A (I don't think the Hermitian-ness...

All matrices A\in\mathbb{C}^{n\times n} have at least one eigenvector z\in\mathbb{C}^n. I'm interested to know what kind of algorithms there are for the purpose of finding an eigenvector. I noticed that \frac{|z^{\dagger} A z|}{\|Az\|} = 1\quad\quad\quad\quad (1) holds only when z is an...

Homework Statement If v and w are eigenvectors with different (nonzero) eigenvalues, prove that they are linearly independent. Homework Equations The Attempt at a Solution Define an operator A such that a is an nxn matrix, and Av=cIv with c an eigenvalue and v and eigenvector. Define a basis...

Homework Statement Suppose that v is an eigenvector of both A and B with corresponding eigenvalues lambda and mui respectively. Show that v is an eigenvector of A+B and of AB and determine the corresponding eigenvalues Homework Equations The Attempt at a Solution Av = lambda*v Bv...

Eigenvalue and eigenvector for a symmetric matrix Homework Statement Let A be a n by n real matrix with the property that the transpose of A equals A. Show that if Ax = lambda x, for some non-zero vector x in C(n) then lambda is real, and the real part of x is an eigenvector of A...

Let A be a nonsingular matrix. What can you say about the convergence of GMRES to the solution of Ax=b when b is an eigenvector of A? I know that if A is a nonsingular matrix with minimal polynomial of degree m, then GMRES solves Ax=b in at most m iterations. Because b is an eigenvector...

Homework Statement For the matrix A = -1, 5 -2, -3 I found the eigenvalues to be -2 + 3i and -2 - 3i. Now I need some help to find the eigenvectors corresponding to each. Homework Equations The Attempt at a Solution For r = -2 + 3i, I plugged that into the (A - Ir) matrix...

To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue. Let's say I already have some method that gives me all the eigenvectors and all...

Homework Statement Directly show that the n x 1 vector [1 1 1 ...1]T is an eigenvector of An. What is its associated eigenvalues? Homework Equations N/A The Attempt at a Solution I started going over this topic since we did not cover it in class due to time constraints and I do not know...

Homework Statement Find the eigenvectors of the following matrix: \[ \left( \begin{array}{ccc} 1 & 1 \\ 4 & -2 \end{array} \right)\] Homework Equations N/A The Attempt at a Solution I already know how to find the solutions. They are {1 1} and {-1 4}. My question is this: could a...

Hi, this is actually for my general relativity class, but I thought I would get more help in the math section of the forums, since it involves very little physics, or even not at all. Homework Statement Take Tab and Sab to be the covariant components of two tensors. Consider the determinant...

ok, i know how to find an eigenvalue and an eigenvector that's fine, what i don't remember is how to normalize your eigenvectors in my problem i have 2 eigenvectors, (1,3) and (3,1) (1,3) corresponds to eigenvalue 10 (3,1) corresponds to eigenvalue 20 in my notes i have written 'to...

Hi, this is my first post here, so bare with me. So I need to compute the eigenvectors of a large matrix (1000x1000) to (10000x10000x) or so. However, I already have the eigenvalues and diagonal/superdiagonal form of the matrix. The equation (A-lambda*I)*v = 0, where A is the matrix, lambda...

Hi, I'm working with stochastic matrices (square matrices where each entry is a probability of moving to a different state in a Markov chain) and I am looking for transforms that would preserve the dominant eigenvector (the "stationary distribution" of the chain). What I want to do is to...

Suppose A\overrightarrow{x}=\lambda_1\overrightarrow{x} A\overrightarrow{y}=\lambda_2\overrightarrow{y} A=A^T Take dot products of the first equation with \overrightarrow{y} and second with \overrightarrow{x} ME 1) (A\overrightarrow{x})\cdot...

Homework Statement The matrix,A,given by A = \left( \begin{array}{ccc} 7 & -4 & 6\\ 2 & 2 & 2 \\ -3 & 4 & -2 \ \end{array} \right) has eigenvalues 1,2,4 . Find a set of corresponding eigenvectors. Hence find the eigenvalues of B, where B = \left( \begin{array}{ccc}...

[SOLVED] Complex Eigenvector I need to solve for an eigenvector using the complex eigenvalue -1 + i \sqrt{11} . I have a matrix: A = \left(\begin{array}{cc}-3 & -5 \\3 & 1\end{array}\right) From the equation A \vec{V} = \lambda \vec{V} , where \vec{V} = (x, y) I get : -3x - 5y =...

Homework Statement Hi, I'm having a bit of a problem normalizing eigenvectors with complex entries. Currently the eigenvector I'm looking at is \[\vec{v}= \left(\begin{array}{c} -2+i\\ 1 \end{array}\right)\] Homework Equations The Attempt at a Solution If the eigenvectors...

Homework Statement The linear operator T on R^2 has the matrix [4 -5; -4 3] relative to the basis { (1,2), (0,1) } Find the eigenvalues of T. Obtain an eigenvector corresponding to each eigenvalue.Homework Equations The Attempt at a Solution I was able to find the eigenvalues (8 and -1)...

Problem : in the Attach file

I'm reading a proof where there's a conclusion: "Since zW\subset W, there is an eigenvector v\neq 0 of z in W, zv=\lambda v." There W is a subspace of some vector space V, and z is a matrix, in fact a member of some solvable Lie algebra \mathfrak{g}\subset\mathfrak{gl}(V). (Could be irrelevant...

http://orion.math.iastate.edu/vika/cal3_files/lec33267.pdf i searched eigenvectors on google and this showed up. here are some problems i need further explaining for example 3 and 4. 3. where do they get e1+e2= 0 equation from? then where did they get e= (1,-1) 4. where do they...

Hi Guys, I have got some enquires for eigenvalue and eigenvector. Consider the 1st matrix: A = [ 1 2 3] [ 0 5 6] [ 0 6 5] The characteristic polynomial is det(A-λI) = [ 1-λ 2 3] [ 0 5-λ 6] [ 0 6...

I need to solve the differential equation \mathbf{x'} = \left( \begin{array}{ccc} 3 & 0 & -1\\ 0 & -3 & -1\\ 0 & 2 & -1 \end{array} \right) \mathbf{x} solving for the eigenvalues by taking the determinate and using the "basketweave" yields (3 - \lambda)(-3-\lambda)(-1-\lambda) +...

The problem is to solve the differential equation where \mathbf{x'} = \left( \begin{array}{cc} 1 & -5\\ 1 & -3 \end{array} \right) \mathbf{x} given that \mathbf{x(0)} = \left( \begin{array}{cc} 5 \\ 4 \end{array} \right) The eigenvalues are easy to find, and they are: \lambda...

If I find a given eigenvector , that vector spans the entire eigenspace defined by that eigenvalue correct? Let's say I get v=(2,1) as an eigenvector. That is the same as saying v=(4,2) right? since they are spanning the same space?

Hey all, I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H? From these pairs of eigenvalue is it possible to obtain the eigenvectors? I don't quite know how to procede any suggestions? Thanks...

Hi Given a 3x3 matrix A = \[ \left[ \begin{array}{ccc} 0 & 0 & 1+2i \\ 0 & 5 & 0 \\ 1-2i & 0 & 4 \end{array} \right] I need to a another 3x3 which satisfacies D = U^-1 A U Step 1. Finding the eigenvalues 0 = det(A- \lambda I ) = (0- \lambda)(\lambda - 5) (\lambda -4...

Lin Alg - Eigenvector Existence proof (More of a proof ?, than eigenvector ?) Here is the question from my book: Show that If \theta \in \mathbb{R}, then the matrix A = \left(\begin{array}{cc}cos \theta & sin \theta \\ sin \theta & -cos \theta \end{array}\right) always has an...

Hi, I've got these two matrices (V & T) and omega square, which is what I have found to be the eigenvalues. Could anyone tell me if this is the way to find the eigenvectors for these matrices and if they are correct? Thanks... http://img305.imageshack.us/img305/6937/ok4zd.jpg

hey, not sure how this works on this website but was just wondering if someone can tell me if I'm doing this right... find characteristic equation of matrix A=l 4 0 1 l l -2 1 0l l -2 0 1l i found it to be (L=lambda)... L^3 - 6L^2 + 11L - 6 = 0 only because i solved...

Hi could someone explain to me how to verify that a vector is an eigenvector of a matrix without explicitly carrying out the calculations which give the eigenvalues of the matrix? Here is an example to illustrate my problem. Q. Let M = \left[ {\begin{array}{*{20}c} { - 3} & 1 & { - 2} & 4...

Hi all, I have the following question. A = nxn non singular matrix I = nxn identity matrix li = eigevalues of A i=1,2...n ui = eigenvectors corresponding to the previous eigenvalues. It true that ( A - l1 * I ) * x =0 is satisfied by any vector of the form x = a1 * u1...

If you have the following kernel (I think that's what it's called): A-\lambda I=\begin{pmatrix}4 & 1 \\ 4 & 1\end{pmatrix} You could write the eigenvector as: \operatorname{span}\begin{pmatrix}1 \\ -4\end{pmatrix} My question is: does it matter how you write the "span" part of it...

Hi I have this here matrix A = \left[ \begin{array}{ccc} 2 & 1 & 0 \\ 0 & 1 & 0 \\ 3 & 3 & 0 \end{array} \right] I calculate the eigenvalues and get (2,1,-1) Next I calculate the eigenvectors and get (1,0,1) and (-1,1,0) and (0,0,0) My professor says my third eigenvector is...

Hi, Does different eigenvector algorithm give different result? eg. using QL with implicit shifts frm (Numerical Recipes) vs Matlab's LAPACK routines? or anyone knows what method Matlab's LAPACK uses & where i can find the source code in c++? Are eigenvectors unique? Thanks!

can anyone explain the the real meaning and purpose of eigen vlaue and eigen vectors.. :smile: