Eigenvectors Definition and 452 Threads

Homework Statement Let A = \left[ \begin{array}{cc} -6 & 0.25 \\ 7 & -3 \end{array} \right] Find an invertible S and a diagonal D such that S^{-1}AS=DHomework Equations ...The Attempt at a Solution So first I need to get eigenvalues so I can get the eigenvectors which will give me the...

dear all how do you find the eigenvalues and eigenvectors of a complex matrix? 0 ; -i ; 0 ; 0 i ; 0 ; -i*sqrt(2) ; 0 0 ; i*sqrt(2) ; 0 ; -i*sqrt(5) 0 ; 0 ...

Homework Statement The matrix is: |1 2| |3 4|The Attempt at a Solution I've worked out the eigenvalues to be \stackrel{\underline{5\pm\sqrt{33}}}{2} But when I plug the first eigenvalue back in I get: |1 - \stackrel{\underline{5+\sqrt{33}}}{2}......2 | |3......4 -...

Homework Statement Let B = (1 1 / -1 1) That is a 2x2 matrix with (1 1) on the first row and (-1 1) on the second. Homework Equations The Attempt at a Solution A) (1 1 / -1 1)(x / y) = L(x / y) L(x / y) - (1 1 / -1 1) (x / y) = (0 / 0) ({L - 1}...

Hi, I'm applying the Lanczos algorithm to find the minimal eigenvalue of some huge matrix. Now that I've got it I'm trying to find the eigenvector corresponding to this eigenvalue. Now I have looked through book after book after book and I have yet to find an explanation of how to do this...

Two questions. First, I'm given a 3x3 matrix with the last row all zeroes. I'm asked to diagonalizable it, but the determinant is 0, so there are no eigenvalues. Am I reasoning correctly here? It seems an odd question to ask. Second, I'm asked to prove that if A n x n matrix in C space, then...

Homework Statement I have matrix A = 4 2 0 1 Whose eigenvalues I found to be 4 & 1 I need to find the eigenvectors for the same matrix Homework Equations (A-lambdaI)V=0The Attempt at a Solution Lambda = 4 gives 0 2 x V1 = 0 0 -3 V2 0v1 + 2v2 = 0 0v1 - 3v2 = 0...

Homework Statement Find the eigenvectors of the matrix A Homework Equations 3. The attempt at the solution \[ \left( \begin{array}{cc}4 & -5 \\1 & 0 \end{array} \right)\] First I find the characteristic equation A - \lambda I \[ \left( \begin{array}{cc}4-\lambda & -5...

Hi, I'm trying to derive the Kullback-Leibler divergence between two multi-variate gaussian distributions, and I need the following property. Is there a simple way to understand this? Prove that: Given that E has orthonormal eigenvectors u_{i} and eigenvalues \lambda_{i} Then: trace(A*E) =...

I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me. "Properties of the eigenvalue when A is symmetric. If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times), orthogonal...

Homework Statement x1(t) and x2(t) are functions of t which are solutions of the system of differential equations x(dot)1 = 4x1 + 3x2 x(dot)2 = -6x1 - 5x2 Express x1(t) and x2(t) in terms of the exponential function, given that x1(0) = 1 and x2(0) = 0 The Attempt at a Solution I've already...

Consider the nXn matrix A whose elements are given by, A_{ij} = 1 if i=j+1 or i=j-1 or i=1,j=n or i=n,j=1 = 0 otherwise What are the eigenvalues and normalized eigenvectors of A??

Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix M from a matrix of its eigenvectors as columns P and a diagonal matrix of the eigenvalues E through M=P E P^{\dagger}?

Eigenvalues & Eigenvectors !SOLVED! Homework Statement Find the eigenvalues and eigenvectors of matrix A = \left( \begin{array}{cc} 2 & 2 \\ 3 & 1 \end{array} \right) Homework Equations Ax = \lambda x The Attempt at a Solution Solving \left\vert \begin{array}{cc} 2 - \lambda &...

Homework Statement Let T: V---->V be a linear operator where dim V=n. Show that V has a basis of eigenvectors if and only if V has a basis B such that TB is diagonal.Homework Equations The Attempt at a Solution Let T=[a1,1...an,1] ai,j=/=0 [a1,n...an,n]...

Hello I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0. I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me? cheers

Hello Im trying to find the eigenvalues and eigenvectors of 3x3 matricies, but when i take the determinant of the char. eqn (A - mI), I get a really horrible polynomial and i don't know how to minipulate it to find my three eigenvalues. Can someone please help.. Thanks

Homework Statement Let A and B be similar matrices a)Prove that A and B have the same eigenvalues Homework Equations None The Attempt at a Solution Firstly, i don't see how this can even be possible unless the matrices are exactly the same :S

Suppose that f is an eigenvector of T with corresponding eigenvalue \lambda. Then f' = T(f) = \lambdaf. This is a first-order differential equation whose solutions are of the form f(t) = ce^(\lambda*t) for some scalar constant c. Consequently, every real number \lambda is an eigenvalue of T...

How can you show that an arbitrary n \times n matrix has n linearly independent eigenvectors? What if all you know about the matrix is that it's the product of a positive-definite matrix and a semi-positive-definite matrix?

If x is an eigenvector of matrix A, is it true that it is also an eigenvector of A -1, or A + A^2? Thanks for the help.

Hi, I am a little confused how do you find out when a matrix has two independent eigenvectors or when it has one or when it has more than two, or is it possible it can have no eigenvectors.

Homework Statement I'm trying to find the unit eigenvectors corresponding to the following matrix A = [5 -2; -2 8] ; means new row Homework Equations det(A - hI) = 0 The Attempt at a Solution I get lambda = 4 and 9 unit eigenvector corresponding to lambda = 4 x1 = ( -2/sqrt(5)...

Hello, please can someone tell me how to decouple and solve this equation? It was on a problem sheet, but the solution jumped to the decoupled equation... =( \frac{dx}{dt} = 2x+y-t \frac{dy}{dt}=2x-y+t I know that it can rewritten as \frac{d}{dt}\left[ \begin{array}{cccc} 2 & 1\\...

When finding eigenvectors in matrices I choose something for some x-es. Like sometimes x3 or x4 is chosen to be s or t or 2s etc... What I´d like to ask about is, does it not matter what the number is? Can I chose whatever I want to? If the matrix has 3 eigenvalues and after gauss...

I don't understand where -4+2y=0 and y=2x comes from Is it obtained after finding the determinant or are the equations reconstructed that the matrix was created from? http://users.on.net/~rohanlal/eigen1.jpg I also don't understand what's happening here. Where did the x(1,2) come from and...

Homework Statement how to find eigenvectors by using gauss jordan A=[1 1; 2 2] Homework Equations I know how to use gauss jordan but don´t know how to use it to find eigenvectors The Attempt at a Solution First I find the eigenvalues: ((y-1)(y-2)-(1*2)=> y1=0 and y2=3 Then I...

Homework Statement Given the characteristic polynomial -2+x-2x^2-x^3, find the eigenvalues and eigenvectors of the matrix [-1, -1, 0] [1, 1, 1] [3, 1, -2] Homework Equations The Attempt at a Solution The eigenvalues are -2.659, 0.329-.802i, and 0.329+.802i. Next you plug each eigenvalue into...

my question is take A= {(5,0,-1),(2,3,-1),(4,0,1)} find all eigenvalues and eigenvectors by using the characteristic equation i get -(lamda-3)3 however its the next bit i don't understand, in the answers (A-3I)(x,y,z)=(0,0,0) is used which I'm perfectly ok with and then (A-3I)2 is used and...

I'm trying to teach myself quantum mechanics from Dirac, and I'm having trouble justifying some of the maths, in particular how we can just jump out of the confines of a Hilbert space when it's convenient. Dirac rather liberally talks about observables that have a continuous range of...

1. Find the eigenvalues and the eigenvectors corresponding to eigenvalues of the matrix A = \left[\begin{array}{ccccc} 1 & 3 \\ 4 & 2 \end{array}\right] 3. The Attempt at a Solution (\lambda I - A) = \lambda \left[\begin{array}{ccccc} 1 & 0 \\ 0 & 1 \end{array}\right] -...

Homework Statement a matrix A: [1 3 0 3 1 0 0 0 -2] Find Q and D where QTAQ=D The Attempt at a Solution I found the eigenvalues of -4,2,2 When I plug them back in and rref the matrix I only get the trivial solution meaning the matrices are linearly independent. How do I get...

revising for a text and got stuck mid way through a question Find the eigenvalues and vectors of A (in matrix form i will state colum then column then column) A((3,0,0),(0,2,0),(0,0,2) B=((3,0,0),(0,2,1),(0,0,2) for A i got x(lamda)=(lamda-3)(lamda-2)^2 lamda=2or3 then i got lamda=3 solved...

I can't follow an argument in Horn and Johnson's Matrix analysis in a suggestion (actually an outline of a proof) that follows problem 8 following section 1.3 (pg 55 in my copy). They argue that if A and B are complex square matrices of order n which commute, and if all eigenvalues of B are...

Homework Statement Suppose that the 2x2 matrix A has eigenvalues lambda = 1,3 with corresponding eigenvectors [2,-1]^T and [3,2]^T. Find a formula for the entries of A^n for any integer n. And then, find A and A^-1 from your formula. Homework Equations Ax = lambda X (P^-1)AP = D A =...

I have come to a problem where I have Eigenvalues = 2,2i,-2i and my Eigenvectors have i's in them. I usually check my work using my calculator to perform the operation of, S^{-1}AS=J where S is my Eigenvector matrix, A is my original. I then see what my J matrix looks like. It should...

I am studying QM by myself. I got a quite confusing problem which annoying me for a certain time. Well, this question is about the angular momentum opeator Lx, Ly and Lz. The matrix form for these operatore are given, so by solving the corresponding secular equation, it is easy to find the...

Homework Statement Hi! I just used MATLAB to find the eigenvalues and eigenvectors of A=[0 -1; 1 0] I obtained the eigenvalues of 0 +/- i and eigenvectors of v(1) = [ 0.7071; 0 - 0.7071i] and v(2) = [ 0.7071; 0 + 0.7071i] Homework Equations I'm having trouble interpreting these results in...

Hi I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A 5 5 5 5 5 5 5 5 5 The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there...

finding the eigenvectors (and behavior of solution) around the critical points found in this thread: https://www.physicsforums.com/showthread.php?t=258349&referrerid=110346 D_{f} = \[\begin{pmatrix}32x & 18y \\ 32x & -32y\end{pmatrix}\] D_{f}(1,1) = \[\begin{pmatrix}32 & 18 \\ 32 &...

Hey Hello, I am dealing with som Principal Component Analysis Can anyone explain why the first eigenvector of a covariance matrix gives the direction of maximum variability. why this special property of eigenvectors

So I understand that if an nxn matrix has n distinct eigenvalues that you can diagonalize the matrix into S\LambdaS^{-1}. This is important because then this form has lots of good properties (easy to raise to powers, etc) So when there are not n distinct eigenvalues, you then solve...

How to get eigenvalues & eigenvectors of N x N matrix? Please can anyone help me out i have searched a lot but not able to find the solution. Regards

This is my last week in Linear Algebra. I am working on our last homework assignment before the exam so I want to make sure I know what I am doing. In each part, make a conjecture about the eigenvectors and eigenvalues of the matrix A corresponding to the given transformation by considering...

I just finished Differential Equations, and I know how to find eigenvalues/eigenvectors, and I understand how to use them to solve a differential equation. But I don't really understand "what they are". How is a matrix with complex eigenvalues any different than a matrix with real...

Homework Statement I have a matrix A = [3 -2; -3 2], then I peform det(A-λI) and find solutions λ=0 and λ=5. For the first case λ=0 I perform A-0I, which of course is just A [3 -2; -3 2]. I then reduce to row echelon form and find the values V1 = 2 and V2 = 3, therefore for λ=0 my...

Homework Statement Show that if an nxn matrix A has n linearly independent eigenvectors, then so does A^T The Attempt at a Solution Well, I understand the following: (1) A is diagonalizable. (2) A = PDP^-1, where P has columns of the independent eigenvectors (3) A is...

Homework Statement Find the eigenvalues and correspointing eigenvectors of the matrix: [1,1;1,1] Homework Equations The Attempt at a Solution I can solve the determinant to get the eigenvalues: e1=2, e2=0, and from here I am supposed to sub these values back into my matrix...

I have two problems here; one I think I almost have but I'm stuck, and the other I'm pretty much stumped on.Homework Statement Suppose V is a complex vector space and T is in L(V). Prove that T has an invariant subspace of dimension j for each j = 1, ... dim(V). Homework Equations The Attempt...

i got this question in which we are given the matrix T and we need to find the eigenvalues and the independent spaces (i don't know what is independent space) of T^2 +2*T the problem is that he started to solve the question as i would have solved it but then he puts a big X on it and does...