Eigenvectors Definition and 462 Threads

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...

Homework Statement Given the matrix 1 1 1 -1 3 1 -1 1 3 x=3 is an eigenvalue and (1,1,1) is a corresponding eigenvector x=2 is an eigenvalue of A of multiplicity 2 Find the eigenvector(s) corresponding to x=2 The Attempt at a Solution (A-AI)= -1 1 1 -1 1 1 -1 1 1...

Hey guys, need some quick help before an exam I have a differential eqn. x' = | 0 1 | *x , and initial conditions x(0) = |2| | -25 10 | |3| I find that there are two eigenvalues 5, and 5 The corresponding eigenvector to 5 is [1 5]...

Homework Statement I am part way done with this problem... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now. W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it. a)...

I guess this is best explained with an example. The matrix (0 -1) has the eigenvalues ------------------------------------------------------------------ (1 0) i and -i. For -i we obtain ix1-x2=0 and x1+ix2=0. I got a corresponding eigen vector (1 i), but when I controlled this result with...

Homework Statement solve the system of first-order linear differential equations: (y1)' = (y1) - 4(y2) (y2)' = 2(y2) using the equation: (λI -A)x = 0 Homework Equations using eigenvectors and eigenvalues in the book 'Elementary Linear Algebra' by Larson and Falvo - Section 7.4 #19...

Find a matrix that has eigenvalues 0,18,-18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1). ... I know the diagonlize rule, and the the rule to find a a power of A A= PDP^-1 D=P^-1AP ... but i am lost as to how to contine... help please?

Homework Statement This is a simple example from the book, but it gets the point across nicely. In this problem eigenanalysis is used as a method to solve linear systems. The matrix... [4 2] [3 -1] Eigenvalues are -2, 5. Homework Equations (A-\lambda I)v=0 x'=[above matrix]x The...

I'm trying to derive the equation for the scalar product of one particle momentum eigenvectors \Psi_{p,\sigma} ( p is the momentum eigenvalue and \sigma represents all other degrees of freedom), in terms of the little group of the Lorentz group with elements W that take the standard four...

Simple Yes or No Will Do... Eigenvectors Homework Statement I think my prof made a mistake when writing this problem: Find a basis of the space V of all 3 x 3 matrices A for which the vectors <1, 1> and <1, 2> are eigenvectors and thus determine the dimensions of V. Is this problem...

Homework Statement Let A= [0 2 1;-2 3 0;1 0 2] Determine a real canonical form of A and give a change of basis matrix P that brings the matrix into this form. Homework Equations The Attempt at a Solution I found my eigenvalues to be 0, 2+i and 2-i. So, taking 2+i, I get the real...

Hey guys, I'm studying some quantum physics at the moment and I'm having some problems with understanding the principles behind the necessary lineair algebra: 1) If two operators do NOT commutate, is it correct to conclude they don't have a similar basis of eigenvectoren? Or are there more...

I have an (unknown) matrix A and with real non-negative values. I know its largest eigenvalue \lambda and the associated eigenvector, v. (I know nothing about the other eigenvectors). Does this give me any information about the eigenvector of AT associated with \lambda or is it completely...

Homework Statement Suppose that a matrix A has real entries (which we always assume) and a complex (non-real) eigenvalue  \lambda= a + ib, with  b not equal to 0. Let W = U + iV be the corresponding complex eigenvector, having real and imaginary parts U and V , respectively. Show that U...

Homework Statement Suppose the the matrix A is symmetric, meaning that A = a b b d Show that for any symmetric matrix A there are always real eigenvalues. Also, show that the eigenvectors corresponding to two dierent eigenvalues are always orthogonal; that is, if V1 and V2 are the...

I remember reading a theorem that said that for an n x n matrix A, there exists a basis of Cn consisting of generalized eigenvectors of A. For A = [1 1 1; 0 1 0; 0 0 1] (the semicolons indicate a new row so that A should be 3 x 3 with a first row consisting of all 1's and a diagonal of 1's)...

Homework Statement Given the matrix A = [1 0 0 -2 1 3 1 1 -1] Find an invertable matrix X and a diagonal matrix D such that A = XDX^-1 Homework Equations A = XDX^-1The Attempt at a Solution I've found that the eigenvalues are -2, 2...

We have two nxn matrices with non-negative elements, A and B. We know the eigenvalues and eigenvectors of A and B. Can we use this information to say anything about the eigenvalues or eigenvectors of C=A*B? The largest eigenvalue of C and the associated eigenvector are of particular interest...

Homework Statement I am given the Hamiltonia operator of a system in two-dimensional Hilbert space: H = i\Delta(|w1><w2| + |w2><w1|) and am asked to find the corresponding eigenstates. I wrote this operator as a matrix, where H11 = 0, H22 = 0, and H12= i\Delta and H21= -i\Delta...

Homework Statement Find the eigenvalues and eigenvectors of this matrix. [4 0 1/2 ] [0 -5 0 ] [1/2 0 1 ] [b]3. The Attempt at a Solution I have found the eigenvalues = -5, 5/2 + sqrt(5/2), 5/2 - sqrt(5/2) I know to get the eigenvectors you subtract the...

Homework Statement So I have to find the eigenvalues and eigenvectors of A=\left(\begin{array}{cc}0&1\\1&0\end{array}\right) which is not that special and hard. I solve the characteristic equation and find the eigenvalues: \lambda_{1,2}=\pm 1. So finding the eigenvectors is relatively...

* corresponds to matrix product I'm working on a method of visualising graphs, and that method uses eigenvector computations. For a certain square matrix K (the entries of which are result of C_transpose*C, therefore K is symmetric) I have to compute the eigenvectors. Since C is mXn, where...

Can someone explain these to me?

I read that n different eigenvalue matrix has always n eigenvectors. But I cannot find any. Here is the state transition function, A: \left[\begin{array}{cc}\dot{I}\\ \dot{U}\end{array}\right] = \left[\begin{array}{cc}0&-1/L\\ 1/C&0\end{array}\right] \left[\begin{array}{cc}{I}\\...

I'm trying to find the Eigenvectors and eigenvalues of this matrix: [ 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 ] I get 0, 1, and -1 as my eigenvalues. Starting with 0, I solve for reduced row echelon form and get the matrix: [ 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 ] My question is, and maybe my...

Homework Statement A = [2,1,2;2,1,2;2,1,2] Find the Eigenvectors of A Homework Equations The Attempt at a Solution First I found the eigenvalues of A det(A - \lambda I) = 0 \lambda = 0,5 __________________________________________ A-5I...

Homework Statement I attached the problem in a picture so its easier to see. Homework Equations The Attempt at a Solution These are the values i got \lambda_ 1 = 1 \lambda_ 2 = -1 x_1 = [-i; 1] (x_1)^H = [i 1] x_2 = [ i; 1] (x_2)^H = [-i 1] * where x_1 and x_2 are...

Find the characteristic polynomial, eigenvectors, and eigenvalues of the matrix. [ 2 -2 3 0 3 -2 0 -1 2 ] My characteristic poly is x^3 - 7x^2 + 14x - 8 The roots are 1, 2, and 4. When solving the system, (2I - A)x = 0 I get: [ 0 1 0 0 0 0 1 0 0 0 0 0 ] Can...

Homework Statement in seeking of eigenvalues and eigenvectors of a given matrix A, is it permissible first to simplify A by means of some elementary operation? (that is, are the eigenvalues and eigenvector of A invariant with respect to elementary row operation)? (prove it)Homework Equations...

I googled for a proof,but didn't find one. Could anyone give me a link to a proof?

how does one systematically find the eigenvectors of a 2x2 (or higher) Real matrix given complex eigenvalues?

The question is at the end of a chapter on spanning vector spaces. Homework Statement Let P denote an invertible n x n matrix. If \lambda is a number, show that E_{\lambda}(PAP^{-1}) = \left\{PX | X\;is\;in\;E_{\lambda}(A)\right\} for each n x n matrix A. [Here E_{\lambda}(A)} is...

Homework Statement For the following linear system: \frac{dx}{dt} = -2x \frac{dy}{dt} = -2y Obtain the general solution. Homework Equations The Attempt at a Solution A= -2 0 0 -2 Using the determinant of A-\lambdaI I got a repeated eigenvalue of -2. I am...

Homework Statement If A=[{5,3},{-2,-2}], find the eigenvectors of A. Using these eigenvectors as matrix P, find P-1 and thus prove P-1AP is diagonal. Homework Equations None The Attempt at a Solution So i can get the eigenvectors to be <3,-1> and <1,-2> corresponding to eigenvalues 4...

Homework Statement Find the eigenvalues and eigenvectors for the matrix [{13,5},{2,4}] Homework Equations None The Attempt at a Solution Well eigenvalues is easy, and turn out to be 14 and 3. So using eigenvalue 3, the two equations 10x1 + 5x2=0 and 2x1 + x2=0. Using these, I assumed...

Homework Statement I have a matrix H= [h g g h] and I need to find the eigenvalues and normalised eigenvectors Homework Equations The Attempt at a Solution I subtracted lamda from the diagonal and then solved for the determinant equally zero. The eigenvalues I found were...

Homework Statement Let P1 and P2 be the projections defined on R^3 by: P1(x1, x2, x3) = (1/2(x1+x3), x2, 1/2(x1+x3)) P2(x1, x2, x3) = (1/2(x1-x3), 0, 1/2(-x1+x3)) a) Let T = 5P1 - 2P2 and determine if T is diagonalizable. b) State the eigenvalues and associated eigenvectors of T...

Homework Statement This isn't really a question in particular. I am doing my first Differential Equations course, and in the complex eigenvalues part, I am getting confused as to how to find the eigenvectors. Example: Solve for the general solution of: x' = (1 -1)x (don't know how to...

I need to compute the 3 eigenvalues and 3 eigenvectors of a symmetric 3x3 matrix, namely a stress tensor, computationaly (in C++). More specific details http://en.wikipedia.org/wiki/Principal_stress#Principal_stresses_and_stress_invariants". Basically 2 questions: 1. I am running into trouble...

If v is an eigenvector of an invertible matrix A, which of the following is/are necessarily true? (1) v is also an eigenvector of 2A (2) v is also an eigenvector of A^2 (3) v is also an eigenvector of A^-1 A) 1 only B) 2 only C) 3 only D) 1 and 3 only E) 1,2 and 3 I am pretty sure...

Homework Statement Determine the eigenvalues and eigenvectors of the matric, A: A=\left[\begin{array}{ccc}1 & 1 & 0\\ 1 & -2 & 0\\ 0 & 0 & 1\end{array} Homework Equations I think i understand what is going on. I have found the matrix equation to be...

Hi all, Let's say we have a symmetric matrix A with its corresponding diagonal matrix D. If A has only 1 eigenvalue, how do we show that there exists 2 eigenvectors? thanks!