Eigenvalues Definition and 820 Threads

Let U, T be linear operators on a vector space V. Prove that UT and TU have the same eigenvalues. Any ideas?

Homework Statement A unitary operator U has the property U(U+)=(U+)U=I [where U+ is U dagger and I is the identity operator] Prove that the eigenvalues of a unitary operator are of the form e^i(a) with a being real. NB: I haven't been taught dirac notation yet. Is there a way i can do...

Homework Statement Consider lowering and rising operators that we encountered in the harmonic oscillator problem. 1. Find the eigenvalues and eigenfunctions of the lowering operator. 2. Does the rising operator have normalizable eigenfunctions?Homework Equations a-= 1/sqrt(2hmw) (ip + mwx) a+...

If after you apply an operator and hence calculate the expectation value of a measureable entity and if you get an eigenvalue, then does that mean when you do the measurement, you will always get the same value for that operator entity, each time? I think yes because otherwise what is so...

Weinberg in volume 1 of his QFT text says we do not observe any non-zero eigenvalues of A = J_2 + K_1; B = -J_1 + K_2. He says the "problem" is that any nonzero eigenvalue leads to a continuum of eigenvalues, generated by performing a spatial rotation about the axis that leaves the standard...

In my calc 3 class, we've taken an alternative(?) route to learning maxes and mins of multivariable equations. By using a Hessian Matrix, we're supposed to be able to find the eigenvalues of a function at the point, and determine whether the point is a max, min, saddle point, or indeterminant...

Suppose that B is the inverse of A. Show that if |psi> is an eigenvector of A with eigenvalue a not equal to 0, then |psi> is an eigenvector of B with eigenvalue 1/a. So I know that A|psi> = a|psi>, and I'm trying to prove that A^(-1)|psi> = 1/a|psi>. I tried simplifying A as a 2x2 matrix...

I'm currently researching a 3d tensor, where certain combinations of terms can cause the principal values (eigenvalues) to become complex. This would then seem to imply that the associated eigenvectors would also become complex. What now, if this tensor were part of a larger equation...

Find the Eigenvalues of the matrix and a corresponding eigenvalue. Check that the eigenvectors associated with the distinct eigenvalues are orthogonal. Find an orthogonal matrix that diagonalizes the matrix. (1)\left(\begin{array}{cc}4&-2\\-2&1\end{array}\right) I found my eigenvalues to...

This is how the book introduced eigenvectors: I do not get how the normal vector of x-y = 0 is <1,-1> . Isn't that saying that the x-component is 1 and the y-component is -1? Also how did they get the vector equation <x,y> = t<1,-1> + <a,b> ? Finally, why does \vec{OQ} = \vec{OP}...

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!

Hey all, I have two matrices A,B which commute than I have to show that these eigenvectors provide a unique classification of the eigenvectors of H? From these pairs of eigenvalue is it possible to obtain the eigenvectors? I don't quite know how to procede any suggestions? Thanks...

Hi again, Question: \hat{A} is an Hermitian Operator. If \hat{A}^{2}=2, find the eigenvalues of \hat{A} So We have: \hat{A}\left|\Psi\right\rangle=a\left|\Psi\right\rangle But I actually don't know how to even begin. \hat{A} is a general Hermitian operator, and I don't know where...

If I have two eigenfunctions of an operator with the same eigenvalue how do I construct linear combinations of my eigenfunctions so that they are orhtogonal? My eigenfunctions are: f=e^(x) and g=e^(-x) and the operator is (d)^2/(dx)^2

Find the eigenvalues of the hamiltonian H=a(S_A \cdot S_B+S_B \cdot S_C+S_C \cdot S_D+S_D \cdot S_A) where S_A, S_B, S_C, S_D are spin 1/2 objects _________________________ I rewrite it as H=(1/2)*a*[(S_A+S_B+S_C+S_D)^2-(S_A+S_C)^2-(S_B+S_D)^2] then i define...

Hey, A Hamiltonian has 3 eigenkets with three eigenvalues 1, sqrt(2) and sqrt(3) - will the expectation values of observables in general be period functions of time for this system? I don't know how to procede? Thnaks in advance

Hey guys, I know its possible to diagonalize a matrix that has repeated eigenvalues, but how is it done? Do you simply just have two identical eigenvectors?? Cheers Brent

I'm trying to create an algorithm in MATLAB, but I have a problem. According to theory, if G is a positive definite matrix, then it's eigenvalues are positive real numbers. I'm using function EIG() to calculate the eigenvalues and eigenvectors of matrices, but I almost always take and negative...

"sum" over Eigenvalues... Is there any mathematical meaning or it's used in Calculus or other 2branch" of mathematics de expression: \sum_{n} e^{-u\lambda (n) } where every "lambda" is just an Eigenvalue of a linear operator: L[y]=-\lambda _{n} y We Physicist know it as the...

This is a very simple question, but I can't seem to get it right, there's probably something silly that I'm missing here. Here's the question: I have A system in the l=1 state, and I have L_z|\ket{lm} = \hbar m\ket{lm}and L^2 \ket{lm} = \hbar^2 l(l+1)\ket{lm} I need to find the eigenvalues...

:rolleyes: :cool: I have a question..yesterday at Wikipedia i heard about the "Hermite Polynomials2 as Eigenfunctions of Fourier (complex?) transform with Eigenvalues i^{n} and i^{-n}...could someone explain what it refers with that?...when it says "Eigenfunctions-values" it refers to the...

I'm having trouble with this: Prove that if P is a linear map from V to V and satisfies P^2 = P, then trace P is a nonnegative integer. I know if I find the eignevalues , their sum equals trace P. But how do I find them here? any thoughts? Thanks

Why do they have to purely imaginary? I got a proof that looks like Ax=ax where a = eigenvalue therefore Ax.x = ax.x = a|x|^2 Ax.x = x.(A^t)x where A^t = transpose = -A x.(-A)x = -b|x|^2 therefore a=-b, where b = conjugate of a Now is this as far as i need to go?

Please help me I am hopelessly lost and don't even know where to start! I guess they're right when they said girls suck at math! It's not fair! :redface: Let A be an nxn matrix with only real eigenvalues. Prove that there is an orthogonal matrix Q such that (Q^T)AQ is upper triangular with...

matrix A = \left(\begin{array}{ccc}3&0&0 \\ 0&3&0 \\3&0&0 \end{array}\right) has two real eigenvalues lambda_1=3 of multiplicity 2, and lambda_2=0 of multiplicity 1. find the eigenspace. A = \left(\begin{array}{ccc}3-3 &0&0 \\ 0&3-3&0 \\3&0&0-3 \end{array}\right) A =...

I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need...

If I have two positive definite Hermitian NxN matrices A and B, if I adiabatically change the components of A to B (constraining any intermediate matrices to be Hermitian as well, but not necessarily positive definite) while \"following\" the eigenvalues ... will the mapping of the eigenvalues...

"Suppose V is a (real or complex) inner product space, and that T:V\rightarrow V is self adjoint. Suppose that there is a vector v with ||v||=1, a scalar \lambda\in F and a real \epsilon >0 such that ||T(v)-\lambda v||<\epsilon. Show that T has an eigenvalue \lambda ' such that |\lambda...

In my textbook recently I stumbled across the following: Give a general description of those matrices which have two real eigenvalues equal in 'size' but opposite in sign? Could anyone explain this again very simply :-)

Hey! Does someone know of some resources which describe how to code a function which calculates the eigenvalues of a matrix? This could be either resources on the net or a book. If you know of a good book which teaches about programming and mathematics together in general I'd be happy to know...

Show that every eigenvalue of A is zero iff A is nilpotent (A^k = 0 for k>=1) i m having trouble with going from right to left (left to right i got) we know that det A = product of the eignevalues = 0 when we solve for the eigenvalues and put hte characteristic polynomial = 0 then det...

I'm asked to find the eigenvalues and eigenvectors of an nxn matrix. Up until now I thought eigenvectors and eigenvalues are something that's related to linear transformations. The said matrix is not one of any linear transformation. What do I do?

is there any trick for finding the eigenvalues and vectors for this kind of matrix? \left( \begin{array}{ccccc} 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & \sqrt{\frac{3}{2} & 0 & 0 \\ 0 & \sqrt{\frac{3}{2} & 0 & \sqrt{\frac{3}{2} & 0 \\ 0 & 0 & \sqrt{\frac{3}{2} & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ \end{array}...

I need help with an integral eigenvalue equation...I am lost on how to handle this: \int_{-\infty}^{\infty} dy K(x,y) \psi_n(y) = \lambda_n \psi_n(x) The kernel, K(x,y) is a 2D, correlated Gaussian. I have read that for this case an analytic solution exist for the eigenvalues, \lambda_n...

I need to find the eigenvalues and eigenvectors of a matrix of the form \left ( \begin{array}{cc} X_1 & X_2 \\ X_2 & X_1 \end{array} \right ) where the X_i's are themselves M \times M matrices of the form X_i = x_i \left ( \begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots &...

if you have a differential equation of the form x' = Ax where A is the coefficient matrix, and you get a triple eigenvalue with a defect of 1. (meaning you get v1 and v2 as the associated eigenvector). How do you get v3 and how do you set up the solutions? I tried finding v3 such that...

Let A be an nxn mx with n distinct eigenvalues and let B be an nxn mx with AB=BA. if X is an eigenvector of A, show that BX is zero or is an eigenvector of A with the same eigenvalue. Conclude that X is also an eigenvector of B. I could show BX is zero or is an eigenvector of A with the...

I've got a homework problem that I am needing to do; however, I am not sure really what the question is asking. Obviously since I don't know what is being asked, I don't know where to begin. I was hoping for some insight. Question: Show that matrix A = {cos (theta) sin (theta), -sin...

I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help. Question: An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must...

I have two excercises which have been causing me to tear my hair off for some time now. (a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k) (b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A (c) the smallest/largest...

Hi, How can you infer from these equations, a = b_{max}(b_{max}+\hbar) \quad \text{and} \quad a = b_{min}(b_{min}-\hbar), that b_{max} = -b_{min}? It is used in the derivation of the angular momentum eigenvalues...

I have the tridiagonal matrix (which comes from the backward Euler scheme) A = [ 1+2M - M 0 ... ] [ -M 1+2M 0 ... ] [ ... ] [ -M 1+2M ] I am given that the...

Hi, I'm wondering if there is some kind of shortcut for finding the eigenvalues and eigenvectors of the following matrix. C = \left[ {\begin{array}{*{20}c} {0.8} & {0.3} \\ {0.3} & {0.7} \\ \end{array}} \right] Solving the equation \det \left( {C - \lambda I} \right) = 0, I...

confused on finding Eigenvalues and Eigenvectors! hello everyone, i can't understand this example, how did they find the Eigen value of 3?! Aslo an Eigen vector of 1 1? http://img438.imageshack.us/img438/1466/lastscan1oc.jpg thanks.

Hi I'm stuck on the following question and I have little idea as to how to proceed. Note: I only know how to calculate eigenvalues of a matrix, I don't many applications of them(apart from finding powers of matrices). Also, I will denote the inner product by <a,b> rather than with circular...

I'm experiencing difficulties trying to find the eigenvalues of the follow matrix. The hint is to use an elementary row operation to simplify C - \lambda I but I can't think of a suitable one to use or figure out whether a single row operation will actually make the calculations simpler. C...

hi, i have trouble understanding these two terms. can anyone explain to me eigenvalues and eigenvectors in laymen terms? Thks in advance! :smile:

howdy all, i need some answers if possible suppose i have a particle mass m, confinded in a 3d box sides L,2L,2L what would be the energy eigenvalues of this particle i presumed it to be: hcross*w*A where hcross is h/2*pi w is omega and A is the...

This is probably a straight forward question, but can someone show me how to solve this problem: \frac {d^2} {d \phi^2} f(\phi) = q f(\phi) I need to solve for f, and the solution indicates the answer is: f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi} I know...