Eigenvalues Definition and 820 Threads

Hello, Given equilibrium, why does one only consider a mixed state where the pure states are eigenfunctions of the hamiltonian, i.e. states with an energy eigenvalue? And for the second question, I quote David Tong's ``Lectures on Statistical Physics'' ( freely and legally accessible on...

I'm not exactly looking for help finding the eigenvalues of the spin operator, I'm mainly wondering if there is a better technique to do it. Homework Statement Find the eigenvalues and corresponding eigenstates of a spin 1/2 particle in an arbitrary direction (θ,\phi) using the Pauli...

For a Jordan form (which is a direct sum of individual jordan blocks) that has distinct eigenvalues along the diagonal, how would the exponential of the Jordan form be calculated ? As far as I am aware the formula for calculating the exponential of a jordan block can only involve one...

Homework Statement I solved a problem that asked me to show that the sum of the eigenvalues of A+B equals the sum of all the individual eigenvalues of A and B, and similarly for products. I just would like to know why is this so... Homework Equations The Attempt at a Solution Is...

Homework Statement The matrix A has 3 distinct eigenvalues t1< t2< t3. Let vi be the unique eigenvector associated to ti with a 1 as its first nonzero component. Let D= [t1 0 0 0 t2 0 0 0 t3] and P= [v1|v2|v3] so that the ith column of P is the eigenvector vi...

Normally when I work in matlab, the eigenvalues of a matrix are arranged in order from least to greatest when I call the function eig(Matrix). But for some reason, Matlab has decided to start arranging them in order of magnitude (greatest to least)... so that it would arrange the following...

helppp please

Hello again everyone! I would like to ask a question regarding this Hamiltonian that I encountered. The form is H = Aa^+a + B(a^+ + a). Then there is this question asking for the eigenvalues and ground state wavefunction in the coordinate basis. The only given conditions are, the commutator...

What's the proof that eigenfunctions of discrete eigenvalues are in Hilbert Space?

Say that for two mxm matrices, they have equivalent eigenvalues If this is the case, is it safe to assume that the Jordan forms of both matrices will be the same ? My reasoning comes from the fact that a general jordan block is represented by the following matrix λ 1 0 0 0 λ 1 0 0 0...

Homework Statement i = the 3x3 matrix below 2-λ 0 1 -1 4-λ -1 -1 2 0-λ Using remainder and factor theorem find the 3 values of λ. Homework Equations |i| = a1|b2c3-c2b3|-a2|a2c3-c2a3|+a3|a2b3-b2a3| |a|=ad-bc The Attempt at a Solution(2-λ) |(4-λ x...

Hi everyone! I am answering this problem which is about the eigenvalues and eigenfunctions of the Hamiltonian given as: H = 5/3(a+a) + 2/3(a^2 + a+^2), where a and a+ are the ladder operators. It was given that a = (x + ip)/√2 and a+ = (x - ip)/√2. Furthermore, x and p satisfies the...

Homework Statement V is a vector space consisting all functions f:R->R that is differentiable many times (a) Let T:V->V be the transformation T(f)=f' Find the (real) eigenvectors and eigenvalues of T (b) Let T be transformation T(f)=f" Prove that all real number, m is the eigenvalue of...

Homework Statement This is the matrix A, which i need to find the eigenvalues and eigenvectors. 3x3 matrix 5 6 12 0 2 0 -1 -2 -2 The attempt at a solution I have found the eigenvalues to be: 1, 2, 2. So, the final eigenvalues are : 1 and 2. Now, i found the eigenvector for...

I know some of their applications, but I wanted to know how they first appeared. Why were eigenvalues and eigenvectors needed?

Homework Statement How does one find all the permissible values of b for -{d\over dx}(-e^{ax}y')-ae^{ax}y=be^{ax}y with boundary conditions y(0)=y(1)=0? Thanks. Homework Equations See aboveThe Attempt at a Solution I assume we have a discrete set of \{b_n\} where they can be regarded as...

Is it possible to diagonalize such matrix? and how would one do it?

The following question was posed on an old qualifying exam for linear algebra: Suppose A is an n by n complex matrix, and that A has spectral radius <1 (the eigenvalue with largest norm has norm <1). Show that A^n approaches 0 as n goes to infinity. The solution is easy when the eigenspace...

Hello i have this matrix \in Z mod 7, M = \begin{pmatrix} 0&6\\ 5&0 \end{pmatrix} always modulo 7 in Z. I found characteristic polynomial x^2+5. Eigenvalues are \lambda = 3, \lambda' = 4 Eigenvectors related to \lambda = 3 are the non-zero solution of the system: 4x +6y = 0, 5x+4y = 0 I...

Homework Statement Prove: If a, b, c, and d are integers such that a+b=c+d, then A=[a b] [c d] has integer eigenvalues, namely,λ_1{}=a+b and λ_2{}=a-c Homework Equations No relevant equation. The Attempt at a Solution No idea :(

The master equation of the damped harmonic oscillator is \frac{d}{dt}\rho_S(t) = -i\omega_0 [a^\dagger a,\rho_S(t)] + \gamma_0(\bar n+1) \{ a\rho_S(t) a^\dagger -\frac{1}{2} a^\dagger a \rho_S(t) -\frac{1}{2} \rho_S(t) a^\dagger a \} + \gamma_0\bar n \{ a^\dagger \rho_S(t) a...

Homework Statement Express y'' + 5y' - 24y = 0 as a system of couple first order DEs, find the eigenvalues of the system and the nature of the critical point at the origin. As well as find the general solution to the system of coupled equations and sketch some trajectories in the phase...

Homework Statement If A is similar to A^(-1) (=inverse of A), must all the eigenvalues equal 1 or -1? Homework Equations The Attempt at a Solution I don't know why the textbook gives me the specific value 1 or -1. If A is similar to its inverse, are the eigenvalues really 1 or...

Homework Statement Find the eigenvalues and eigenvectors of the matrix A = [2, 1; 8, 4] Homework Equations det(A - lambda I) = 0 The Attempt at a Solution After expanding using the formula I have the equation (2 - \lambda)(4 - \lambda) - 8 Which gives \lambda = 0, 6 (Should I...

I'm reading from Wikipedia: I thought linear operators always had eigenvalues, since you could always form a characteristic equation for the corresponding matrix and solve it? Is that not the case? Are there linear operators that don't have eigenvalues?

Ive been trying for 3 hours now and can't seem to find the eigenvalues, the long polynomials are getting me confused, the matrix is [2 2 1:1 3 1:1 2 2] So far i did [2-L 2 1:1 3-L 1:1 2 2-L] then I do the normal way to find the determinant but after that I get a horrible polynomial...

Homework Statement I think my teacher made a mistake in his homework answer. I need to verify this for practice. The answer I got is below. The answer the teacher has is in the pdf. Homework Equations Please refer to attached pdf The Attempt at a Solution So there is two...

Hi, While trying to solve an optimization problem for a MIMO linear precoder, I have encountered the need to compute the eigenvalues of a matrix D^{H}A^{H}AD where the matrix A is known and the matrix D is a diagonal matrix whose entries contain the variables that need to be optimized (those...

Homework Statement find the eignevalues (a part of a larger problem) for A= | -4 1 1 | | 1 5 -1 | | 0 1 -3 | Homework Equations The Attempt at a Solution = | -4-x 1 1 | | 1 5-x -1...

I'm looking for a proof of the fact that orthogonal eigenfunctions of a Hermitian operator have distinct eigenvalues. I know the proof the converse: that eigenfunctions belonging to distinct eigenvalues are orthogonal. thanks alot!

i have been trying to learn a bit of quantum mechanics,this is some thing that has been bothering me , if the states of a system can be expressed as vectors in the Hilbert space,what is the motivation behind saying that physical observables can be given by operators?even then ,how can we say...

Homework Statement Find the eigenvalues/vectors of A. (I can do this bit :P, A is a 3x3 matrix) What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix? Homework Equations The Attempt at a Solution I have (correctly) found that A...

Hi there, Iam just wondering that at different values of m and n, the position of eigenvalues are always varies accordingly. I mean, the outputs positions of eigenvalues are not consistent given by mathematica. Please see attached file for reference. My question is that how to fix this...

How do i find the eigenvalues of cos(x) -sin(x) sin(x) cos(x) and cos(x) sin(x) sin(x) -cos(x) Thanks

I have a matrix A, which contains only positive real elements. A is a differentiable function of t. Are the eigenvalues of A differentiable by t?

Hi, Can anyone help me prove that two commuting matrices can be simultaneously diagonalized? I can prove the case where all the eigenvalues are distinct but I'm stumped when it comes to repeated eigenvalues. I came across this proof online but I am not sure how B'_{ab}=0 implies that B is...

Homework Statement Let A=LU and B=UL, where U is an upper triangular matrix and L is a lower triangular matrix. Demonstrate that A and B have the same eigenvalues. Homework Equations Not sure. The Attempt at a Solution I know that if I can show that A and B are similar (so if I...

Thanks, although I still haven't managed to factorise the expression although I did type it up in LaTeX! Homework Statement Prove by induction that the following statement is true for all positive integers n. If \lambda is an Eigenvalue of the square matrix A, then \lambda^n is an eigenvalue...

Homework Statement find the general solution to x'=Ax; where A is a 3x3matrix: A=[0 1 1; 1 0 1; 1 1 0] Homework Equations det(A-lambda*I)=0 The Attempt at a Solution i found the eigenvalues to be 2, -1, -1. for lambda=2 i found the corresponding eigenvector to be a 3x1 martrix...

Given a square matrix, if an eigenvalue is zero, is the matrix invertible? I am inclined to say it will not be invertible, since if one were to do singular value decomposition of a matrix, we would have a diagonal matrix as part of the decomposition, and this diagonal matrix would have 0 as an...

Homework Statement I don't know how to put matrices in, so I'll just link an http://forum.bodybuilding.com/attachment.php?attachmentid=3339921&d=1305058219" Basically find the solution for that matrix. Homework Equations The Attempt at a Solution This was the...

Let L : V>>>V be an invertible linear operator and let lambda be an eigenvalue of L with associated eigenvector x. a) Show that 1/lambda is an eigenvalue of L^-1 with associated eigenvector x. For this question, the things I know are that L is onto and one to one. Therefore, how to prove this...

I have a 3 x 3 matrix A = (0 -1 -3) (2 3 3) (-2 1 1) Let & represent lambda here. I am trying to find the eigenvalues of A. I start off by taking the characteristic equation of A and end up with -&[(&-3)(&-1) -3] + (2& - 8) - 3(-2& + 8) yet can't then get that factored down...

Homework Statement When generating a matrix from eigenvectors, does it matter in which order the columns are placed?

Homework Statement a.) The motion of a particle in the 3-dimensional space is described by the Hamiltonian H = Hx+Hy+Hz, where Hx=1/2*(px2+x2), Hy=1/2*(py2+y2), Hz=1/2*(pz2+z2) Check that the standard angular momentum operators Lx, is a constant of motion. b.) By knowing that the...

x1' = x1 - 5x2 x2' = x1 + 3x2 \begin{bmatrix} 1 & -5\\ 1 & 3\end{bmatrix} \begin{bmatrix} 1-\lambda & -5\\1 & 3-\lambda\end{bmatrix} The eigenvalue I have is lambda = 2+/- 2i. Using lambda = 2-2i, I get the following: \begin{bmatrix} -1+2i & -5\\1 & 1+2i\end{bmatrix} I get an...

I am trying to get an eigenvector for the following matrix, I am up to the final step. 4 1 0 0 I got it to be -1 4 is this the same as 1 -4 sorry I am pretty new to linear algebra.

Homework Statement A 3x3 matrix with all 9 of the numbers being .3 Find all the eigenvalues. Homework Equations The Attempt at a Solution I worked through it and I ended up with (l=lamda) l^3-.9l^2+.54l-.162=0 With my calculator I found one of the values, which means that there...

Hi, I'm new in this forum. I have a problem i can't solve and searching on Google i couldn't find anything. It says: If D(g) is a representation of a finite group of order n , show that K = \sum^{i=1}_{n} D^{\dagger} (g_i) D(g_i) has the properties: b) All eigenvalues of K are...

Homework Statement http://img703.imageshack.us/img703/4489/unledzh.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Ax = λx Ax = x Ax - x = 0 (A - I)x = 0 I set up my matrix...