Eigenvectors Definition and 462 Threads

I thought I understood how to solve these sorts of equations, but apparently not.. 1. Homework Statement In Linear Algebra I'm solving diff eqs with eigenvectors to get all the combinations that will solve for a diff eq. The text then asked me to check my answer by going back and solving...

Extremely confused on finding eigenvectors? Below I have a picture that gives the matrice and the eigenvectors. How did the solution find these eigenvectors?? i.e. the eigenvalues are 7 and -2 IMAGE LINKS http://tinypic.com/r/2liii68/9 http://tinypic.com/view.php?pic=2liii68&s=9#.VkY_YfmrSUk

I know how to solve \frac{d\vec{u}}{dt} = A\vec{u}, I was just watching a lecture, and the lecturer related that solving that equation is pretty much a direct analogy to \vec{u} = e^{At}\vec{u}(0), in so far as all we need to do after that is understand exactly what it means to take the...

So, in a section on applying Eigenvectors to Differential Equations (what a jump in the learning curve), I've encountered e^{At} \vec{u}(0) = \vec{u}(t) as a solution to certain differential equations, if we are considering the trial substitution y = e^{\lambda t} and solving for constant...

MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable". imgur: http://i.imgur.com/Q4pbi33.jpg and imgur: http://i.imgur.com/RSOmS2o.jpg Upon rereading...I do see the possibility...

Homework Statement find eigenvalues and eigenvectors for the following matrix |a 1 0| |1 a 1| |0 1 a| Homework EquationsThe Attempt at a Solution I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3 - a...

I am looking for a clear example on how to obtain the complete set of eigenvalues and eigenvectors for a dense, non-hermitian matrix using cuSolver.

What does it mean when it says eigenvalues of Matrix (3x3) A are the square roots of the eigenvalues of Matrix (3x3) B and the eigenvectors are the same for A and B?

Hi there. How would I show that the eigenvalues of a matrix are an invariant, that is, that they depend only on the linear function the matrix represents and not on the choice of basis vectors. Show also that the eigenvectors of a matrix are not an invariant. Explain why the dependence of the...

Don't exist formula for the eigenvectors, all right!? Eigenvectors needs be found manually, correct!? But and about the Mohr's circle? This physical/mathematical theory don't define clearly the direction of the eigenvectors (called principal direction) with the eigenvalues (called principal...

So, I have the matrix: A = -1 -3 3 9 Eigenvalues i calculated to be λ = 8 and 0 Now when i calculate the Eigenvector for λ = 8, i get the answer -1 3 Then when solve for...

Hi Folks, I calculate the eigenvalues of \begin{bmatrix}\cos \theta& \sin \theta \\ - \sin \theta & \cos \theta \end{bmatrix} to be \lambda_1=e^{i \theta} and \lambda_2=e^{-i \theta} for \lambda_1=e^{i \theta}=\cos \theta + i \sin \theta I calculate the eigenvector via A \lambda = \lambda V as...

I calculate 1) ##\Omega= \begin{bmatrix} 1 & 3 &1 \\ 0 & 2 &0 \\ 0& 1 & 4 \end{bmatrix}## as not Hermtian since ##\Omega\ne\Omega^{\dagger}## where##\Omega^{\dagger}=(\Omega^T)^*## 2) ##\Omega\Omega^{T}\ne I## implies eigenvectors are not orthogonal. Is this correct?

say for example when I calculate an eigenvector for a particular eigenvalue and get something like \begin{bmatrix} 1\\ \frac{1}{3} \end{bmatrix} but the answers on the book are \begin{bmatrix} 3\\ 1 \end{bmatrix} Would my answers still be considered correct?

I have the following matrix: [-1 3 -3(2)^0.5 ; 3 -1 -3(2)^0.5 ; -3(2)^0.5 -3(2)^0.5 2] I was able to find the eigenvalues as -4 and 8. I am now trying to find the corresponding eigenvectors how since -4 is a double root i am unsure how to go about this. I have tried using gaussian...

Homework Statement Look at the matrix: A = sin t sin p s_x + sin t sin p s_y +cos t s_z where s_i are the pauli matrices a) Find the eigenvalues and normalized eigenvectors (are they orthogonal)? b) Write the eigenvector of s_x with positive eigenvalue as a linear combination of the...

Mod note: I revised the code below slightly, changing the loop control variable i to either j or k. The reason for this is that the browser mistakes the letter i in brackets for the BBCode italics tag, which causes some array expressions to partially disappear. Hello, I am trying for the first...

Homework Statement I have 4 equations. 3x+6y-5z-t=-8 6x-2y+3z+2t=13 4x-3y-z-3t=-1 5x+6y-3z+4t=-6 I have already solved this matrix using gauss elimination and found that x=1, y=2, z=5, t=-2 Now the next part of the question asks to solve the matrix using eigenvalues and eigenvectors...

Consider the system: $x' = x + y + z$ $y' = 0x + 2y + 3z$ $z' = 0x + 0y + 3z$ a)Find the eigenvalues for the systemSo after doing my $3 \times 3$ matrix I got: $\lambda_1 = -3$, $\lambda_2 = 1$, and $\lambda_3 = 2$ , is this correct? b)Find an eigenvector for the smallest eigenvalue So I am...

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of **A**. e) Find eigenvectors associated with each eigenvalue. Indicate which eigenvector goes with which eigenvalue. f)...

Homework Statement A linear transformation with Matrix A = ## \begin{pmatrix} 5&4&2\\ 4&5&2\\ 2&2&2 \end{pmatrix} ## has eigenvalues 1 and 10. Find two linearly independent eigenvectors corresponding to the eigenvalue 1. Homework Equations 3. The Attempt at a Solution [/B] I know from the...

Hi, I'k looking at some MATLAB code specifically eig2image.m at: http://www.mathworks.com/matlabcentral/fileexchange/24409-hessian-based-frangi-vesselness-filter/content/FrangiFilter2D So, I understand how the computations are done with respect to the eigenvector / eigenvalues and using...

Homework Statement I think this problem is supposed to be pretty simple, but I have almost no knowledge of how to use matlab. I was told to use this function: [V,D]=eig(A) to give me the eigenvectors (columns of matrix V) and the diagonal matrix with eigenvales in the diagonal ( matrix D). I...

Hi, I've written a little fortran code that computes the three Eigenvectors \vec{v}_1, \vec{v}_2, \vec{v}_3 of the inertia tensor of a N-Particle system. Now I observed something that I cannot explain analytically: Assume the position vector \vec{r}_i of each particle to be given with respect...

I have a bit of problem with zero eigenvectors and zero eigenvalues. On one hand, there seems to be nothing in the definition that forbids them, and they even seem necessary to allow because an eigenvalue can serve as a measurement and zero can be a measurement, and if there is a zero eigenvalue...

From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...

From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...

Homework Statement I know that Unitary operators act similar to hermitean operators. I want to prove that the eigenvalues of unitary operators are complex numbers of modulus 1, and that Unitary operators produce orthogonal eigenvectors. Homework Equations U†U = I U-1=U† λ = eiΦ{/SUP]...

Homework Statement Find the eigenvectors and eigenvalues of exp(iπσx/2) where σx is the x pauli matrix: 10 01 Homework Equations I know that σxn = σx for odd n I also know that σxn is for even n: 01 10 I also know that the exponential of a matrix is defined as Σ(1/n!)xn where the sum runs...

Suppose you have two observables ##\xi## and ##\eta## so that ##[\xi,\eta]=0##, i know that there exists a simultaneous complete set of eigenvectors which make my two observables diagonal. Now the question is, if ##\xi## is a degenerate observable the complete set of eigenvectors still exist?

Homework Statement Let V be a finite dimensional vector space over ℂ . Show that any linear transformation T:V→V has at least one eigenvalue λ and an associated eigenvector v. Homework EquationsThe Attempt at a Solution Hey everyone I've been doing sample questions in the build up to an exam...

Homework Statement Consider the initial value problem for the system of first-order differential equations y_1' = -2y_2+1, y_1(0)=2 y_2' = -8y_1+2, y_2(0)=-1 If the matrix [ 0 -2 -8 0 ] has eigenvalues and eigenvectors L_1= -4 V_1= [ 1...

all the eigenvectors of a matrix are perpendicular, ie. at right angles to each other, HOW? I can imagine three eigenvectors as three perpendicular axes. How can be more than three axes are perpendicular with respect to each other?

Homework Statement I try to run this program, but there are still some errors, please help me to solve this problems Homework EquationsThe Attempt at a Solution Program Main !==================================================================== ! eigenvalues and eigenvectors of a real...

Hi, friends! In order to find an orthogonal basis of eigenvectors of the Fourier transform operator ##F : L_2(\mathbb{R})\to L_2(\mathbb{R}),f\mapsto\lim_{N\to\infty}\int_{[-N,N]}f(x)e^{-i\lambda x}d\mu_x## for Euclidean separable space ##L_2(\mathbb{R})##, so that ##F## would be represented by...

Hello, I am currently trying to study the mathematics of quantum mechanics. Today I cam across the theorem that says that a Hermitian matrix of dimensionality ##n## will always have ##n## independent eigenvectors/eigenvalues. And my goal is to prove this. I haven't taken any linear algebra...

Hello everyone, I've got the eigenvectors of a matrix H (Hessenberg matrix) obtained from the decomposition A=QHQ'. Now I seek the eigenvectors of the matrix A. I've found somewhere that it should be : eigenvector_of_A=Q*eigenvector_of_H but some numerical test with MATLAB doesn't agree. For...

Statement: I can prove that if I apply a function to my matrix (lets call it) "A"...whatever that function does on A, it will do the same thing to the eigenvalues (I can prove this with a similarity transformation I think), so long as the function is basically a linear combination of the powers...

Homework Statement Find the eigenvalues and normalized eigenfuctions of the following Hermitian operator \hat{F}=\alpha\hat{p}+\beta\hat{x} Homework Equations In general: ##\hat{Q}\psi_i = q_i\psi_i## The Attempt at a Solution I'm a little confused here, so for example I don't know...

Hi, I have an issue with zeroing the 3x3 matrix to find the eigenvector. I have found the characteristic equation for the 3 eigenvalues. the matrix is 1 1/2 1/3 1/2 1/3 1/4 1/3 1/4 1/5 The equation i got is -A^3 + (23/15)A^2 - (127/720)A + (1/2160) which...

Homework Statement Compute, using wolfram is an option, eitH where H is: ##H = \begin{pmatrix} 0 & -1 & 0 & 0 & 0& 0\\ -1 & 0 & -1 & 0 & 0& 0\\ 0 & -1 & 0 & -1 & 0& 0\\ 0 & 0 & -1 & 0 & -1& 0\\ 0 & 0 & 0 & -1 & 0& -1\\ 0 & 0 & 0 & 0 & -1& 0\\ \end{pmatrix}## Homework...

I have a non symmetric matrix AB where A and B are symmetric matrices. How can I find the eigenvectors and eigenvalues of AB? In a paper( Fisher Linear Discriminant Analysis by M Welling), the author asks to find eigenvalues and eigenvectors of B^(1/2)* A *B^(1/2) which is a symmetric...

I've got a problem which is asking for the eigenvalues and eigenstates of the Hamiltonian H_0=-B_0(a_1 \sigma_z^{(1)}+a_2 \sigma_z^{(2)}) for a system consisting of two spin half particles in the magnetic field \vec{B}=B_0 \hat z . But I think the problem is wrong and no eigenstate and...

Hi, folks I have had a hard time to find out whether or not there is a theorem in Linear Algebra or Spectral Theory that makes any strong statement about the relationship between the entries of a Matrix and its Eigenvalues and Eigenvectors. Indeed, I would like to know how is the...

Homework Statement I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of A + kI. The Attempt at a Solution ## A\vec{v}=\lambda\vec{v} ## ## (A+kI)\vec{v}=\lambda\vec{v} ## ## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} -...

Homework Statement I am asked to find the diagonal matrix of eigenvalues, D, and the matrix of corresponding eigenvectors, P, of the following matrix: \begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & -2\\ 0 & 0 & -1 \end{pmatrix} Homework Equations The Attempt at a Solution We just started this topic...

Homework Statement Find the Eigenvalues of A= 4 0 1 -2 1 0 -2 0 1 Then find the eigenvectors corresponding to each of the eigenvalues. Homework Equations The Attempt at a Solution I found the Characteristic Polynomial of the matrix, computed the Eigenvalues which are...

Homework Statement The Matrix A is as follows A= [4 -4 0 2 -2 0 -2 5 3] and has 3 distinct eigenvalues λ1<λ2<λ3 Let Vi be the unique eigenvector associated with λi with a 1 as its first nonzero component. Let D = [ λ1 0 0 0 λ2 0 0 0 λ3] and P=...

Hello, I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one Homework Statement A= 2 1 0 1 0 3 -1 0 0 1 1 0 0 -1 0 3 λ1=2 multiplicity 3 λ2=3 multiplicity 1...

Given a vector ##\vec{r} = x \hat{x} + y \hat{y}## is possbile to write it as ##\vec{r} = r \hat{r}## being ##r = \sqrt{x^2+y^2}## and ##\hat{r} = \cos(\theta) \hat{x} + \sin(\theta) \hat{y}##. Speaking about matrices now, the the eigenvalues are like the modulus of a vector and the eigenvectors...