Eigenvalue Definition and 382 Threads

Homework Statement Show that the matrix A = [cos θ -sin θ sin θ cos θ] will have complex eigenvalues if θ is not a multiple of π. Give a geometric interpretation of this result. Homework Equations Ax = λx, so det(A-λI) = 0 The Attempt at a Solution In this case...

So I understand the idea of eigenvalues, eigenvectors, and eigenfunctions corresponding to a given operator on some vector or function space. But I'm just wondering, why are eigenvalues so important in quantum mechanics and physics in general? What I mean is, why are scaled multiples of a...

Say we have an eigenvalue \lambda and corresponding eigenvectors of the form (x,x,2x)^T. What is the geometric multiplicity?

Homework Statement The problem amounts to finding the eigenvalues of the matrix |0 1 0| |0 0 1| |1 0 0| (I have no idea how to set up a matrix in the latex format, if anyone can tell me that'd be great) Homework Equations The characteristic equation for this matrix is...

I'm tinkering with a code snippet where a part finds eigenvalues. eig(A); The thing is, I tried to do it not using eig() to grasp this and got stuck. Could anyone shed some light on this..? How do I find the smallest eigenvalue?

Homework Statement An electron moves in a one-dimensional lattice with the separation between adjacent atoms being equal to a. a. Write down the momentum eigenvalue equation for the electron. b. Find the general form of the solutions of the eigenvalue equation. c. By requiring that the...

Homework Statement Proof: Prove that if A is an nxn (square mtx) such that A^2=A, then A has 0 or 1 as an eigenvalue. The Attempt at a Solution A=A^2 A^2-A=0 A(A-I)=0 A=0 or A=1 and then plugging the A solutions into the characteristic equation and solving for λ

In my lecture notes my prof used the eigenvalue c= 1 + i and ended up with the matrix with (5 3+i) as row 1, and the second row is zeroes. After that, he simply wrote that the basis for this eigenvalue c is (3+i,-5) (in column form) without explaining. How did he get that basis? I tried working...

I have a generalised eigenvalue problem of the form A\boldsymbol{u} = \lambda B\boldsymbol{u}\;, where A and B are symmetric matrices with real symbolic entries. I'm trying to compute the eigenvalues with Mathematica using the command Eigenvalues[{A,B}] which according to the documentation...

Hi guys I have a problem that I need some help with, I am looking for a direct eigenvalue solver algorithm. The problem is that all the eigenvalue solvers I can find seems to reorder the final matrix after the size of the eigenvalues. The matrix it shall calculate is very small (5-10)...

I'm trying to do something that requires solving an eigenvalue problem of the form A_{imkl} c_m c_k c^*_l=\lambda c_i where A is a known rank-4 tensor, \lambda is the eigenvalue, and the c_i's are a set of unknown coefficients that I need to determine. I would guess that this type of problem...

Q)Different eigenvectors corresponding to an eigenvalue of a matrix must be linearly dependant? Is the above statement true or false.Give reasons.

Hey there, first timer poster here Homework Statement I'm working on a barotropic linear instability analysis and I've been having trouble getting an expression for the complex phase speed eigenvalue C = C_r + i*C_i for the purpose of plotting a dispersion diagram (C_i vs k or C_r vs...

Homework Statement Consider the matrix A=[a d f; 0 b e; 0 0 c], where all elements are real numbers (a) what condition(s) on the elements of A are sufficient to guarantee that A has 3 distinct eigenvalues? (b) prove that any two eigenvectors x1 and x2 associated with two distinct...

A is a simetric metrices nxn. so v\in R^n and v\neq 0 so (\lambda I -A)^2=0 for some \lambda prove that for the same v (\lambda I -A)=0 how i tried to solve it: i just collected data from the given. simetric matrices is diagonizable. B=(\lambda I -A) we were given that B^2v=0 so...

Homework Statement This is an example from Gasiorowicz's Quantum Physics. "Example 3-1" is a particle in an infinite potential-well, but that should not matter. Homework Equations The Attempt at a Solution Why is P(-2) (which I suppose is the probability that the eigenvalue -2 is...

Okay, I know that if I can't get n linearly independent eigenvectors out of a matrix A (∈ℝnxn), it is not diagonalizable (and that some necessary conditions for diagonalizability in this regard may be being symmetric and/or having distinct eigenvalues.) This is how things are for the usual...

Dear all, in these http://pages.unibas.ch/comphys/comphys/TEACH/SS04/course.pdf" lecture notes, the author says on page (0-120): http://img15.imageshack.us/img15/615/capturena.png It is not obvious to me, why due to the translation invariance of the energy 3 eigenvalues of the D_IJ matrix have...

Homework Statement Please take a look at the attachment for the problem statement. Homework Equations For 1 dim Harmonic oscillator, E = (n+1/2)h.omega/2pi I don't know which equation to use for 2 dim The Attempt at a Solution I am unable to solve because I don't know which...

Could someone please walk me through answering this question: it looks easy but i forgot how to do it -__- i did A - lamda I = 0, substituted lamda = 5, then my matrix became 0 4 4 0 -2 -2 0 -2 -2 so i put in the form of A|b where b was a zero vector and i row reduced, getting 0 1 1 | 0 0...

Homework Statement a) |-1 1 1| | 1 -1 1| = A | 1 1 -1| Find an orthoginal matrix P that diagonalizes Ab) |0 1| What value of a is multiplicity 2, what value of a is eigen values -1 and 2 A = |a 1| what value of a does A have real eigenvaluesC) If A is a...

I have a rather basic question about solving eigenvalue problems. Once you actually find all the eigenvalues for a given operator in some basis and you go about finding the respective eigenvectors through the components and run into a situation like this: \mid \omega = 1 > \Rightarrow...

Hello everyone, please help me to answer this question. Is this true that any LTI system can be characterized by either its impulse response or engenvalue?

1. Homework Statement Suppose that A is a square matrix and the sum of the entries of each row is some number k. Is k an eigenvalue of A? if so, what is the corresponding Eigenvector?2. Homework Equations Ax-λx=0 3. The Attempt at a Solution (1-k)(K-λ)-k=0I am not sure how to solve this...

I am working on a problem and before I post the remaining questions on it, I want to be sure I calculated the eigenvector correctly. The eigenvalue I used was lambda = 3-4i. \begin{bmatrix} 3-lambda & -4\\ 4 & 3-lambda\end{bmatrix} After substituting, the eigenvector I came up with is V...

Homework Statement Let A be an invertible matrix. show that if λ is an eigenvalue of A, then 1/λ is an eigenvalue of A^-1 PLEASE HELP ME . Thank you.

Homework Statement The question says for the hamiltonian \hat{}H+\hat{}H1 calculate the complete energy eigenvalue spectrum. for the ground state show that the result agrees with the one found by the perturbation theory previously. I'd assume \hat{}H here is just the standard...

So I'm working out on a potentials of the type x^2p, and I have a program that solves and gives the eigenenergies for a potential that I have (x^n in general). I noticed that for a ground state the potential x^4 has the smallest eigenvalue : 0.667981 in units where \hbar=m=\omega=1. I...

I'm taking the math subject GRE in just over a year's time... and I was wondering if there are "ideal" algorithms to have in our tool box to do a computation like this quickly. Obviously the type of matrices in a standardized exam are going to be fairly clean or look dirty but have some less...

We are doing Eigenvalue problems in my Differential Equations class and I just want to make sure I understand some of these concepts. If anyone could look through my current understanding and guide me in the right direction that would be great! So when you have some equation L[y]=\lambda...

Homework Statement For a material the stress is defined by the means of the stress matrix O O = (6 1 -2 1 2 2 -2 2 5) Expressed in MPA It can be derived that the principe stress are: O1= 4-sqrt(13), O2= 5 and O3=4+sqrt(13) I know you can derive the principal...

Homework Statement Solve the differential equation eigenvalue problem: f'' + \lambda f = 0, \quad 0 \leq x \leq 4\pi, \quad \text{where} \quad f^{'}(0) =0, \quad f^{'}(4\pi) = 0, \quad \text{and} f \neq 0. Consider ONLY \quad \lambda \geq 0, \quad and find the values of \quad \lambda...

Assuming that k\geq0, How does one prove that when A has an eigenvaule \lambda that A^{k} has an eigenvalue \lambda^{k}?

I have a PDE test next week and I'm kinda confused. How do you prove that eigenvalues are all positive? I know Rayleigh Quotient shows the eigenvalues are greater than or equal to zero, but can someone explain the next step. Thanks in advance

Dear frnds, suppose one have a 3D matrix, A=ones(3,3,3); he wants to have eigen value of A, then, A-Lamda*I=0 where A is 3D matrix, I is 3D matrix. now Problem is our mathematical soft can only do 2D inversion or eigen calculation. Please refer me any info to find out the...

[b]1. Homework Statement [/ from the ets general physics practice test (ill take it in april) the state of spin 1/2 particles using the eigenstates up and down Sz up= 1/2 hbar Sz down= -1/2 hbar Homework Equations given sigmax (pauli spin matrix) which of the following list...

Does anybody know what physical and mathematical conditions must be in place if we are to represent the states of physical systems using eigenvalue equations (so that we can represent the energy of a state |n> of a particle by H|n>=E|n>, etc?) Thanks!

Homework Statement In quantum mechanics a physical observable is represented by an operator A. Define the terms eigenstate, eigenvalue and eigenfunction of a quantum mechanical operator. Homework Equations The Attempt at a Solution I think I know in that eq 'f' is the eigenfunction, and...

Homework Statement This is a general question... I can easily go from a matrix A to its eigenvalues and then eigenvectors but how would I go from the eigenvalues and eigenvectors to a feasible original matrix? Any thoughts appreciated!

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...

I am new to quantum physics. Is there is any operator in quantum physics that does not show eigenvalue property? Give example please, thankx...

Hi, this is probably really basic for anyone really good with MAPLE but I just solved an Eigenvalue problem in MAPLE and it displays the answer for lambda as a list since my problem contained a 6x6 matrix. My problem is that I want to be able to perform an operation of each individual output...

Homework Statement I am having some trouble with this procedure and I am not exactly sure how to phrase my questions; so I will procede with one particular problem that is giving me trouble and perhaps someone can help to shed light on it. :smile: In one problem, I am given some matrix...

Homework Statement \begin{bmatrix} -7 && -16 && 4\\ 6 && 13 && -2\\ 12 && 16 && 1 \end{bmatrix} Diagonalize the matrix (if possible), given that one eigenvalue is 5, and that one eigenvector is {-2, 1, 2}Homework Equations A=PDP^{-1} The Attempt at a Solution If I were allowed to simply...

Let V=C[x]10 be the fector space of polynomials over C of degree less than 10 and let D:V\rightarrowV be the linear map defined by D(f)=f' where f' denotes the derivatige. Show that D11=0 and deduce that 0 is the only eigenvalue of D. find a basis for the generalized eigenspaces V1(0), V2(0)...

Homework Statement given the following functions: y(x)= Acos(kx) y(x)=A sin(kx)-Acos(kx) y(x)=Acos(kx)+iAsin(kx) y(x)=A d(x-x0) Which are eigenfunctions of the position, momentum, potential energy,kinetic energy, hamiltonian, and total energy operators Homework Equations y(x) is...

Homework Statement A spin system with only 2 possible states H = (^{E1}_{0} ^{0}_{E2}) with eigenstates \vec{\varphi_{1}} = (^{1}_{0}) and \vec{\varphi_{2}} = (^{0}_{1}) and eigenvalues E1 and E2. Verify this & how do these eigenstates evolve in time? Homework Equations...

Homework Statement Homework Equations The Attempt at a Solution I think I have to use the fact that [a+ , a] = 1 but I don't know where to apply this.

Homework Statement Prove or Disprove: a 3x3 matrix A can have 0 as a eigenvalue Homework Equations (xI-A)=0 The Attempt at a Solution I believe it's false just because I've never seen it. I have no idea how to prove it.

Assume A=f(t) is an nxn matrix with all elements being non-negative. The eigenvalue with the largest absolute value is real and positive. We will call it r. Is there an analytical way to calculate dr/dt, or at least find the values of t for which dr/dt=0?