Eigenvalues Definition and 853 Threads

Hey! :o We have the Sturm-Liouville problem $\displaystyle{Lu=\lambda u}$. I am looking at the following proof that the eigenvalues are real and that the eigenfunctions are orthogonal and I have some questions... $\displaystyle{Lu_i=\lambda_iu_i}$ $\displaystyle{Lu_j=\lambda_ju_j...

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 Multiply the matrices to find the resultant transformation. $$x\prime =2x+5y\\ y'=x+3y $$ and $$ x\prime \prime =x\prime -2y\prime \\ y\prime \prime =3x\prime -5y\prime $$ Homework Equations $$Mr=r\prime$$ The Attempt at a Solution I get imaginary eigenvalues of -i and...

If I define a state ket in the traditional way, Say: $$|\Psi \rangle =\sum _{ i }^{ }{ a_{ i }|\varphi _{ i }\rangle \quad } $$ Where $$a_i$$ is the probability amplitude. How does: $$\hat {H } |\Psi \rangle =E|\Psi \rangle $$ if the states of $$\Psi$$ could possibly represent states...

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 From Mary Boas' "Mathematical Methods in the Physical Science" 3rd Edition Chapter 3 Sec 11 Problem 33 ( 3.11.33 ). Find the eigenvalues and the eigenvectors of the real symmetric matrix. $$M=\begin{pmatrix} A & H \\ H & B \end{pmatrix}$$ Show the eigenvalues are real and...

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...

[b]1. Find the corresponding Eigenline A= (3 -3 2 -4) Homework Equations A=(a b c d) k2-(a+d)k+(ad-bc)=0 The Attempt at a Solution k2-(3-(-4))k+(3(-4)-(-3)2)=0 k2+k-6=0 (k+2)(k-3) So k=-2 and k=3 Eigenvector for k=3 (3 -3 2 -4)(x y) = 3(x y) (3x -3y 2x -4y)= (3x...

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=...

Hey! :o $$y''+\lambda y =0$$ $$y(0)=0$$ $$y'(0)=\frac{y'(1)}{2}$$ I have to show that the eigenvalues are complex and are given by the relation $\cos{\sqrt{\lambda}}=2$ except from one that is real. The characteristic equation is $m^2+\lambda =0 \Rightarrow m= \pm \sqrt{ \lambda}$$*$...

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...

Hi everybody.. How can i use fortran99 to find the eigenvalues & eigenvectors of sparse matrices? Thanx :)

Homework Statement A = 7 -5 0 -5 7 0 0 0 -6 Can you please show your method aswell. Every time I try I get the wrong answer. FYI Eigen values are 12.2,-6The Attempt at a Solution so far I got: det = 7-λ -5 0 -5 7-λ 0 0 0 -6-λ Im unsure what to do next. I tried doing...

I have a question on the algebra involved in bra-ket notation, eigenvalues of \hat{J}_{z}, \hat{J}^{2} and the ladder operators \hat{J}_{\pm} The question has asked me to neglect terms from O(ε^{4}) I am using the following eigenvalue, eigenfunction results, where ljm\rangle is a...

Hi, I need to learn the following proof and I'm having trouble getting my head round it. Any help would be appreciated. Show that if vector x in R^n with components x=(x1,x2,...,xn), then x.Lx=0.5 sum(Aij(xi-xj)^2) where A is the graphs adjacency matrix, L is laplacian. Then use this result to...

We know that all eigenvalues of a Hermitian matrix are real. How to explain this from the physics point of view?

Hi all - working on this problem wanted to see if anyone had any advice - thanks! As shown in section 4.4, the poles of the system H(z) with state matrices \mathbf{A, b, c^t, } d are given by the eigenvalues of \mathbf{A}. Find: Show that, if d\neq0, the zeros of the system are given by the...

Homework Statement The Hamiltonian for a two level system is given: H=a(|1><1|-|2><2|+|1><2|+|2><1|) where 'a' is a number with the dimentions of energy. Find the energy eigenvalues and the corresponding eigenkets (as a combination of |1> and |2>). Homework Equations...

Homework Statement Suppose the angular wavefunction is ##\propto (\sqrt{2} cos(\theta) + sin (\theta) e^{-i\phi} - sin (\theta) e^{i\phi})##, find possible results of measurement of: (a) ##\hat {L^2}## (b)##\hat {L_z}## and their respective probabilities. Homework Equations...

Homework Statement For the following wave functions: ψ_{x}=xf(r) ψ_{y}=yf(f) ψ_{z}=zf(f) show, by explicit calculation, that they are eigenfunctions of Lx,Ly,Lz respectively, as well as of L^2, and find their corresponding eigenvalues. Homework Equations I used...

Homework Statement \hat T = \frac{{\hat L_z^2}}{{2I}} = - \frac{{{\hbar ^2}}}{{2I}}\frac{{{\partial ^2}}}{{\partial {\varphi ^2}}} Homework Equations Find eigenfunctions and eigenvalues of this operatorThe Attempt at a Solution It leads to the differential eqn - \frac{{{\hbar...

Homework Statement Is it necessary to arrange the eigenvalues in increasing value order? As shown in the image attached, if I arrange my eigenvalues -2, -1, 1 diagonally, my D would be 2^8 , 1, 1 diagonally. However if i arrange it as, say, -1, 1, -2, my D would be different...

Homework Statement Part (a): Find the eigenvalues and eigenvectors of matrix A: \left( \begin{array}{cc} 2 & 0 & -1\\ 0 & 2 & -1\\ -1 & -1 & 3 \\ \end{array} \right) Part(b): Find the eigenvalues and eigenvectors of matrix ##B = e^{3A} + 5I##. Homework Equations The Attempt at a Solution...

Hi, I am studying quantum mechanics right now and I can't able to understand some questions about Eigenvalues and Eigenvectors. 1. What does the eigenvalue tell us about the quantum mechanical operators i.e. if we operate a momentum operator on ψ what does the Eigen value of that equation...

Hi all, I have two matrices A=0 0 1 0 0 0 0 1 a b a b c d c d and B=0 0 0 0 0 0 0 0 0 0 a b 0 0 c d I need to prove that two eigenvalues of A and two eigenvalues of B are equal. I tried to take the determinant of A-λI...

I am going through Friedberg and came up with a rather difficult problem I can't seem to resolve. If ## F = ℝ ## and A is a normal matrix with real eigenvalues, then does it follow that A is diagonalizable? If not, can I find a counterexample? I'm trying to find a counterexample, by...

Hi All, My question is more from applied quantum mechanics. Suppose I have a 2D conductor(or semiconductor). I use eigenstate representation of hamiltonian in transverse direction and real space representation in longitudinal direction (direction of current flow). Now, 1. Hω=Eω , ω being...

Homework Statement I need to understand how I would go about using QR decomposition of a matrix to find the matrix's eigenvalues. I know how to find the factorization, just stuck on how I would use that factorization to find the eigenvalues. Homework Equations A=QR where Q is an...

Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...

Find the complex eignevalues of the first derivative operator d/dx subject to the single boundary condition X(0) = X(1). So this has to do with PDEs and separation of variables: I get to the point of using the BC and I am left with an expression: 1 = eλ, this is where my issue falls...

Hi, Homework Statement How may I determine whether a system is stable if its input is equal to its output, hence yielding a system(transfer) function equal to 1? Furthermore, could an eigenvalue zero characterize a stable system? I am attaching three examples where I am asked to determine...

Homework Statement ##A=\begin{bmatrix} 16 &{-6}\\39 &{-14} \end{bmatrix}## Homework Equations The Attempt at a Solution I did ##A=\begin{bmatrix} 16-\lambda &{-6}\\39 &{-14-\lambda} \end{bmatrix}## and got that ##\lambda_1=1+3i## and ##\lambda_2=1-3i## The solution...

Suppose A is a diagonlizable nxn matrix where 1 and -1 are the only eigenvalues (algebraic multiplicity is not given). Compute A^2. The only thing I could think to do with this question is set A=PD(P^-1) (definition of a diagonalizable matrix) and then A^2=(PD(P^-1))(PD(P^-1))=P(D^2)(P^-1)...

Hello, I want to generate a (large) matrix with eigenvalues that are all in a small interval. The relationship between the maximum eigenvalue and minimum eigenvalue should be as small as possible, that's the goal. And the eigenvalues must all be positive. Is there any simple way to do...

1. What are the possible eigenvalues of the spin operator \vec{S} for a spin 1/2 particle? Homework Equations I think these are correct: \vec{S} = \frac{\hbar}{2} ( \sigma_x + \sigma_y + \sigma_z ) \sigma_x = \left(\begin{array}{cc}0 & 1\\1 & 0\end{array}\right),\quad...

Homework Statement Given system: dx/dt=-x-5y dy/dt=x+y Homework Equations The Attempt at a Solution So I calculated that \lambda_1=-2i and \lambda_2=2i Generaly \lambda=+-qi next i know that general solution is in form: x=C1cos(qt)+C2sin(qt) y=C*1cos(qt)+C*2sin(qt) So...

Homework Statement Find the eigenvalues of the following Hamiltonian. Ĥ = ħwâ^{†}â + \alpha(â + â^{†}) , \alpha \in |RHomework Equations â|\phi_{n}>=\sqrt{n}|\phi_{n-1}> â^{†}|\phi_{n}>=\sqrt{n+1}|\phi_{n+1}> The Attempt at a Solution By applying the Hamiltonian to a random state n I...

Hi everyone, Just an easy question that came to my mind while studying basics of SUSY. Consider in N=1, D=4 a massive clifford vacuum |m,s,s_3\rangle, and for cconcreteness take its spin to be s=1/2. Now, acting with the four supercharges on both the |m,1/2,1/2\rangle and |m,1/2,-1/2\rangle...

Hello guys, is there any way someone can explain to me in resume what eigen values and eigenvectors are because I don't really recall this theme from linear algebra, and I'm not getting intuition on where does Fourier transform comes from. my teacher wrote: A\overline{v} = λ\overline{v}...

Homework Statement Was given a matrix To find the eigenvalues I set up the characteristic equation [-1-x | 7 | -5 ] [-4 | 11-x | -6 ] [-4 | 8 | -3-x] With some dirty work I got this bad boy out, which I'm having trouble factoring -x3+7x2-15x+9Homework Equations...

The det. of the following matrix: $$ \begin{matrix} 2k-ω^{2}m_{1} & -k\\ -k & k-ω^{2}m_{2}\\ \end{matrix} $$ must be equal to 0 for there to be a non-trivial solution to the equation: $$(k - ω^{2}m)x =0$$ Where m is the mass matrix: $$ \begin{matrix} m_{1} & 0\\ 0& m_{2}\\...

Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...

Q: Using Dirac notation, show that if A is an observable associated with the operator A then the eigenvalues of A^2 are real and positive. Ans: I know how to prove hermitian operators eigenvalues are real: A ket(n) = an ket(n) bra(n) A ket(n) = an bra(n) ket(n) = an [bra(n) A ket(n)]* =...

Hi everyone, :) Here's a question I got stuck. Hope you can shed some light on it. :) Of course if we write the matrix of the linear transformation we get, \[A^{t}.A=\begin{pmatrix}a_1^2 & a_{1}a_2 & \cdots & a_{1}a_{n}\\a_2 a_1 & a_2^2 &\cdots & a_{2}a_{n}\\.&.&\cdots&.\\.&.&\cdots&.\\a_n...

I've been wrestling with this question for a while and can't seem to find anything in my notes that will help me. Homework Statement Determine whether the wave function \Psi (x,t)= \textrm{exp}(-i(kx+\omega t)) is an eigenfunction of the operators for total energy and x component of momentum...

Homework Statement Calculate the eigenvalues of the L_x^2 matrix. Calculate the eigenvalues of the L_z^2 matrix. Compare these and comment on the result. Homework Equations L_x=\frac{1}{2}(L_+ + L_- ) The Attempt at a Solution I have derived eigenvalues for each: 0 and \hbar^2...

Following Butkov, a second order ode A(x)y'' + B(x)y' + C(x)y = D(x) can always be brought into Sturm-Liouville form \tfrac{d}{dx}[p(x)y'] - s(x)y = f(x) after multiplying across by H(x) = - \tfrac{1}{A(x)}e^{\int^x \tfrac{B(t)}{A(t)}dt}. He then says the function s(x) can...

Hi everyone, I have two matrices A and B, A=[0 0 1 0; 0 0 0 1; a b a b; c d c d] and B=[0 0 0 0; 0 0 0 0; 0 0 a b; 0 0 c d]. I have to proves theoretically that two of the eigenvalues of A and B are equal and remaining two eigenvalues of A are 1,1. I tried it by calculating the...

Problem: Let $A$ be a $n \times n$ matrix with real entries. Prove that if $A$ is symmetric, that is $A = A^T$ then all eigenvalues of $A$ are real. Solution: I'm definitely not seeing how to approach this problem. I know that to calculate the eigenvalues of a matrix I need to solve $\text{det...

Standard Pauli spin matrices are: Sx: $$\left(\begin{array}{cc}0&1/2\\1/2&0\end{array}\right)$$ Sz: $$\left(\begin{array}{cc}1/2&0\\0&-1/2\end{array}\right)$$ The Sz eigenvectors are Z+ = (x=1,y=0) and Z- = (x=0,y=1). These yield eigenvalues 1/2 and -1/2 respectively. Similarly...