Eigenvalues Definition and 853 Threads

Homework Statement Let \lambda_0 \in \mathbb{C} be an eingenvalue of the n \times n matrix A with algebraic multiplicity m , that is, is an m-nth zero of \det{A-\lambda I} . Consider the perturbed matrix A+ \epsilon B , where |\epsilon | \ll 1 and B is any n \times n matrix...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

Take a look at this theorem. Is it a way to show this theorem? I would like to show it using the standard way of diagonalizing a matrix. I mean if P = [v1 v2] and D = [lambda1 0 0 lambda D] We have that AP = PD even for complex eigenvectors and eigenvalues. But the P matrix...

Homework Statement I'm trying to demonstrate that if: $$\hat{L}_z | l, m \rangle = m \hbar | l, m \rangle$$ Then $$\hat{L}_z^2 | l, m \rangle = m^2 \hbar^2 | l, m \rangle$$Homework Equations $$\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2$$ $$\hat{L}_z = -i\hbar \left [ x...

OK. An example I have has me stumped temporarily. I'm tired. General spin matrix can be written as Sn(hat) = hbar/2 [cosθ e-i∅sinθ] ...... [[ei∅sinθ cosθ] giving 2 eigenvectors (note these are column matrices) I up arrow > = [cos (θ/2)] .....[ei∅sin(θ/2)] Idown arrow> =...

Suppose we have a matrix A that has eigenvalues λ1, λ2, λ3,... Matrix B is a matrix that has "very small" matrix elements. Then we could expect that the eigenvalues of sum matrix A + B would be very close to the eigenvalues λi. But this is not the case. The eigenvalues of a matrix are not...

hi, please tell me what are the limitations for finding eigenvalues ? thanks

Homework Statement For the conic, 5x2+4xy+5y2=9, find the direction of the principal axes, sketch the curve. I found the eigenvalues as 3,7 but have no idea whether the 'new' equation is 3(x')2+7(y')2 or 7(x')2+3(y')2 is there a way to determine which 'way' it goes? I took a guess...

Hello Physics Forums community, I'm afraid I really need a hand in understanding Why are the Fourier Series for continuous and periodic signals using diferent notation of the Fourier Series for discrete and periodic Signals. I have been following the book " Signals and Systems " by Alan V...

Two questions: If you have two states which have at least one common eigenvalue, then are the two states distinguishable? If you have one state but measure it with two different bases, can one conclude anything if the two measurements have a common eigenvalue? Thanks

Homework Statement We have a three mass two strings system with: m_1 string M string m_2 The end masses are not attached to anything but the springs, the system is at rest, and k is equal for both strings and m_1 and m_2 are equal. The distance between to m_1 and m_2, on both sides of M...

Homework Statement Prove that if A is an nxn matrix with eigenvector v, then v is an eigenvector for Ak where kε(all positive integers) Homework Equations Av=λv The Attempt at a Solution Av=λv A(Av)=A(λv) Akv=λ(Av) i know i may not be doing it right but this is what i can...

Homework Statement Let A and C be nxn matrices with C invertible. Prove that A and C-1AC have the same eigenvalues. Homework Equations B=C-1AC The Attempt at a Solution det(A-λI) =det(B-λI) det(A-λI) =det(C-1AC - λI) det(A-λI) =det(C-1AC - λC-1IC) det(A-λI) =det[CC-1(A-λI)]...

Hi, In the study of dynamical systems, phase portraits play an important role. However, in almost all related text, I only see some standard examples like prey-predator, pendulum etc. I have a rather unclear thought in my head regarding the role of real/imaginary eigenvalues in the system...

Here is the question: Here is a link to the question: Matrix Question?? a.b.c.d.? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could...

Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right...

Homework Statement Given matrix A= {[39/25,48/25],[48/25,11/25]} find the basis for both eigenvalues. Homework Equations The Attempt at a Solution I row reduced the matrix and found both eigenvalues. I found λ = -1, and λ = 3. Then, I used diagonalization method [-1I2 - A 0]...

Could someone please explain to me (with an example if possible) what is the Geometric Multiplicity of Eigenvalues? I cannot understand it from what I have read on the web till now. Thanks in advance.

can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?

Homework Statement Given the endomorphism ϕ in ##\mathbb{E}^4## such that: ϕ(x,y,z,t)=(4x-3z+3t, 4y-3x-3t,-z+t,z-t) find: A)ker(ϕ) B)Im(ϕ) C)eigenvalues and multiplicities D)eigenspaces E)is ϕ self-adjoint or not? explain The Attempt at a Solution I get the associated matrix...

Homework Statement I've tried to solve the following exercise, but I don't have the solutions and I'm a bit uncertain about result. Could someone please tell if it's correct? Given the endomorphism ##\phi## in ##\mathbb{E}^4## such that: ##\phi(x,y,z,t)=(x+y+t,x+2y,z,x+z+2t)## find: A) ##...

Homework Statement Find the eigenvalues and eigenvectors of P = {(0.8 0.6), (0.2 0.4)}. Express {(1), (0)} and {(0), (1)} as sums of eigenvectors. Homework Equations Row ops and det(P - λI) = 0. The Attempt at a Solution I've found the eigenvectors and eigenvalues of P to be 1...

[b]1. The 3x3 Matrix A=[33, -12, -70; 0, 1, 0; 14, -6, -30] has three distinct eigenvalues, λ1<λ2<λ3. Find each eigenvalue.[b]2. det(A-λI)=0 where I denotes the appropriate identity matrix (3x3 in this case)[b]3. Here's my attempt: --> det([33, -12, -70; 0, 1, 0; 14, -6, -30]-λ[1, 0, 0; 0, 1...

Homework Statement The Hamiltonian for a rigid rotator which is confined to rotatei n the xy plane is \begin{equation} H=-\frac{\hbar}{2I}\frac{\delta^{2}}{\delta\phi^{2}} \end{equation} where the angle $\phi$ specifies the orientation of the body and $I$ is the moment of inertia...

Homework Statement We only briefly mentioned this in class and now its on our problem set... Show that all eigenvalues i of a Unitary operator are pure phases. Suppose M is a Hermitian operator. Show that e^iM is a Unitary operator. Homework Equations The Attempt at a Solution...

the problem stays to find the values of Lambda for which the given problem has nontrivial solutions. Also to determine the corresponding nontrivial eigenfunctions. y''-2y'+\lambday=0 0<x<\pi, y(0)=0, y(\pi)=0 r^{2}-2r=-\lambda r=1±i\sqrt{\lambda+1}...

Hi everyone, I am currently dealing with a nonlinear system of coupled equations. In fact I had performed a perturbation approach for this system which is highly nonlinear. Thanks to first step of the perturbative approach I could reach eigenvalues in the "linear case". Right now I want to...

Homework Statement Find the eigenvalues | 1 2 -1| | -5 7 -5 | | -9 8 -7| Homework Equations The Attempt at a Solution I know that i need to add a -λ to every term in the trace so my matrix becomes | 1-λ 2 -1| | -5 7-λ -5| | -9 8 -7-λ| Then i need to...

How come a square matrix has eigenvalues of 0 and the trace of the matrix? Is there any other proof other than just solving det(A-λI)=0?

Homework Statement We have the following conic formula ##ax^2 + 2bxy + cy^2 + dx + ey = ## constant which corresponds to a ellipse, hyperbola or parabola. The second order terms of the corresponding PDE $$ a\frac{\partial^2 u}{\partial x_1^2} + 2b\frac{\partial^2 u}{\partial x_1\partial x_2} +...

I'm new to QM, but I've had a linear algebra course before. However I've never dealt with vector spaces having infinite dimension (as far as I remember). My QM professor said "the eigenvalues of the position operator don't exist". I've googled "eigenvalues of position operator", checked into...

Homework Statement X'=AXA=\left[\begin{matrix} 0 & 1 & 0 \\ -1 & 0 &0 \\0 & 0 & -1\end{matrix}\right] Homework Equations n/a The Attempt at a Solution The eigenvalues are -1, and \pm i. I also can see that the matrix A is already in the form A=\left[\begin{matrix} \alpha & \beta & 0 \\...

Hey there, the question I'm working on is written below:- Let |a'> and |a''> be eigenstates of a Hermitian operator A with eigenvalues a' and a'' respectively. (a'≠a'') The Hamiltonian operator is given by: H = |a'>∂<a''| + |a''>∂<a'| where ∂ is just a real number. Write down the eigenstates...

Homework Statement If the only eigenvalue is zero, can you ever get a set of n linearly independent vectors? Homework Equations The Attempt at a Solution

Homework Statement Define a matrix function f(T) of an nxn matrix T by its Taylor series f(T)=f0 +f1T +f2T2+... Show that if matrix T has the eigenvalues t1,t2...tn, then f(T) has eigenvalues f(t1), f(t2)...f(tn) Homework Equations The Attempt at a Solution I am at a loss of how...

1. In the many statements of the QM postulates that I've seen, it says that if you measure an observable (such as position) with a continuous spectrum of eigenvalues, on a state such as then the result will be one of the eigenvalues x, and the state vector will collapse to the...

Hi dear friends I have a 12*12 symbolic matrix in terms of x y z d that I want its eigenvalues but not mathematica nor MATLAB can do it for me.My mathematica is "7" so If you have a newer version or even in MATLAB , would you mind checking my matrix in your software? this is my matrix in...

Hello, I have a 3D COMSOL model which I am using for the purpose of vibration analysis. Up to now I've got analytical eigenvalues using COMSOL with MATLAB. I have to correlate the results with the measured eigenvalues results, using MAC criterion. The problem is that my analytical results...

Here is the question: Here is a link to the question: Help finding the eigenvalues of a matrix? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement Using eigenvalues and eigenvectors, find the general solution to dx/dt = x - y dy/dt = x + yHomework Equations Matrix 'A' - lambda*identity matrix ; for finding eigenvalues and thus eigenvectors Other relevant equations written on the attached scanned image of my attempt at...

Homework Statement Indicate which of the following expressions yield eigenvalue equations and identify the eigenvalue. a) d/dx (sin(∏x/2)) b) -i*hbar * ∂/∂x (sin(∏x/2)) c) ∂/∂x (e-x^2) The Attempt at a Solution I know that if the wave equation yields an eigenvalue equation, it will...

Hello I was trying to find eigenvalues of a matrix. I calculated the characteristic polynomial by calculating (A-lambdaI) and then calculating it's determinant. The results was: -\lambda ^{3}+8\lambda ^{2}-20\lambda +16 which is the correct calculation. Now, the eigenvalues are 2,2,4, but I...

Hello everyone! I'm curious to know what is the significance of the Eigenvalues of a covariance matrix. I'm not interested to find an answer in terms of PCA (as you of you may be familiar with the term). I'm thinking of a Gaussian vector, whose variance represent some notion of power or...

Homework Statement Find the eigenvalues of the following and the eigenvelctor which corresponds to the smallest eigenvalue Homework Equations I know how to find the eigenvalues and eigenvectors of a 2x2 matric but this one I'm not so sure so any help would be appreciated The...

I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?

I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?

eigenvalues of a compact positive definite operator! Let A be a compact positive definite operator on Hilbert space H. Let ψ1,...ψn be an orthonormal set in H. How to show that <Aψ1,ψ1>+...+<Aψn,ψn> ≤ λ1(A)+...+λn(A), where λ1≥λ2≥λ3≥... be the eigenvalues of A in decreasing order. Can...

Eigen values of a complex symmetric matrix which is NOT a hermitian are not always real. I want to formulate conditions for which eigen values of a complex symmetric matrix (which is not hermitian) are real.

I have always been quite confused about the fact that any measurement MUST yield a real number. What says it must so? Don't we modify our measurement apparatus to yield something which is consistent with the theory. So coulnd't we just imagine having complex values for momentum and position. All...