Eigenvalues Definition and 853 Threads

Homework Statement Solve the following systems by either substitution or elimination: dx/dt = y dy/dt = -x + cos(2t) Homework Equations I know the solution is: x(t) = c_1cos(t) + c_2sin(t) - 1/3cos(2t) y(t) = -c_1sin(t) + c_2cos(t) + 2/3sin(2t) The Attempt at a Solution x' = [ 0 1; -1...

Homework Statement Prove that the eigenvalues of a Hermitian matrix is real. http://www.proofwiki.org/wiki/Hermitian_Matrix_has_Real_Eigenvalues The website says that "By Product with Conjugate Transpose Matrix is Hermitian, v*v is Hermitian. " where v* is the conjugate transpose of v...

$$ \mathcal{J} = \begin{pmatrix} -\sigma & \sigma & 0\\ 1 & -1 & -\sqrt{b(r - 1)}\\ \sqrt{b(r - 1)} & \sqrt{b(r - 1)} & - b \end{pmatrix} $$ From a quick try and error, I was able to find that when $r = 1.3456171$ we will have 3 negative eigenvalues. But when $r = 1.3456172$, there will be a...

Homework Statement If I have a Hamiltonian matrix, \mathcal{H}, that only depends on a kinetic energy operator, do the energy eigenvalues have to be non-negative? I have an \mathcal{H} like this, and some of its eigenvalues are negative, so I was wondering if they have any physical...

Homework Statement Solve X' = [ [9, 4, 0], [-6, -1, 0], [6, 4, 3]] * X using eigenvalues. Homework Equations (A - λI) * K = 0 X = eλt The Attempt at a Solution Set up the characteristic equation to find eigenvalues. I found a root of multiplicity 2 of λ=3 and another distinct root...

Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...

Matrix A= 2 1 2 1 2 -2 2 -2 -1 It's known that it has eigenvalues d1=-3, d2=d3=3Because it has 3 eigenvalues, it should have 3 linearly independent eigenvectors, right? I tried to solve it on paper and got only 1 linearly independent vector from d1=-3 and 1 from d2=d3=3. The method I used...

In Quantum, I ran across the eigenvalue problem. They gave me a matrix, and i was asked to find eigenvalues and then eigenvectors. But the eigenvalues, were degenerate and thus i couldn't find the exact normalized eigenvector. What to do in this case? Shoukd i choose arbitrary values? My...

Hi We have a matrix A (picture), the eigenvalues are λ1 = 4 and λ2 = 1 and the eigenvectors are λ1 : t(1,0,1) λ2 : t1(1,0,2) + t2(0,1,0) I have to examine if there's a column vector v that satifies : A*v = 2 v I would say no there doesn't exist such a column vector v because 2 isn't an...

Homework Statement I'm trying to find the eigenvalues/eigenvectors of the unitary matrix A = (1/√5){{1,2},{2i,-i}} Homework Equations det[A-λI]=0 AA* = I (where A* denotes the adjoint of A)The Attempt at a Solution I have tried to do this straightforwardly, as I would any other matrix, by...

Homework Statement Let \vec{x} and \vec{v} be vectors in \mathbb{R}^3. If A is a matrix such that A\vec{x} gives the projection of \vec{x} onto \vec{v}, then what are the eigenvalues of A and what are their algebraic multiplicities? Homework Equations Eigenvalue: A real number λ is an...

Homework Statement The Attempt at a Solution I know the general form should be x1(t)=-C1sin(3t) + C2cos(3t) x2(t)=C1sin(3t) + C2cos(3t) but there's something going on with v that I'm not getting. I'm not sure how to incorporate it without knowing A

hi friends i want to find eigenvalues of a 4*4 matrix but niether MATLAB nor MATHEMATICA can't solve it. Both of programs said that eiganvalues of matrix is too complicated and have infinite sentences. now what can i do?is there anyway that simplify the steps for MATLAB or mathematica...

Homework Statement Let T be a square nxn matrix and let each entry tij=n(i-1)+j. Calculate the eigenvalues of T of T when T is a 3x3 matrix and a 4x4 matrix. Homework Equations The Attempt at a Solution For the homework I'm supposed to calculate the 3x3 case and the 4x4 case. Then it says...

Alright... So I'm in an 'introductory' Q.M class in college right now, it's the only one that this two-year college has, so I don't have an upper division Q.M Profs to talk to about this, and since my prof is equally confused, I turn to the internet. Okay, so everyone knows that <ψ|Aψ> = <a>...

Let's say I'm looking at the infinite square well. Typically, given some arbitrary initial (normalized) wavefunction, we can decompose it into a linear combination of components of the complete set (on the interval [-a,a] or whatever) of sin's and cos's. Then, if you measure something like the...

Homework Statement Given the matrix A= 1 -1 -1 -1 1 -1 -1 -1 1 Find the eigenvalues.Homework Equations I = identity matrix The Attempt at a Solution det(A-xI) = (1-x)3 - 2 - 3*(1-x) = 0 ⇔ ⇔(1-x)3 - 3*(1-x) - 2 = 0 I can't find a way to solve this equation... Any help...

Like the title says, why are the only possible values of an operator its eigenvalues? reading shankar right now and I'm having difficulty understanding why this has to be the case, given some operator/variable Ω

I am trying to investigate the statistical variance of the eigenvalues of sample covariance matrices using Matlab. To clarify, each sample covariance matrix, \hat{\mathbb{R}}_{nn}, is constructed from a finite number, N, of vector snapshots, each sized (L_{vec} \times 1) (afflicted with random...

hello, Let T be a open, connected and bounded subset of ℝ3 which has a smooth boundary bd(T). Consider the equation Δu = -λu with either the Dirichlet condition (u=0 in bd(T)) or Neumann (where δu/δn = 0 on bd(T)). Define: \left\langle g,h \right\rangle =\iiint_{T}{g(x)h^{*}(x)}dx...

Suppose we have three hermitian operators A,B,C each with only non degenerate eigenvalues. If A and B commute, then for each eigenvector of A we can find an eigenvector of B, and because the eigenvalues are non degenerate the mapping is one to one. If B and C commute we can do the same. This...

Homework Statement Let C = 2,0,-2 1,1,2 -1,-1,-1 Use the Cayley-Hamilton theorem to compute C^3.Homework Equations Cayley-Hamilton theorem says that every square matrix satisfies its own characteristic equation. C^3=PD^3P^{-1} where P is the matrix formed from linearly independant...

Homework Statement y'' +λy=0 y(1)+y'(1)=0 Show that y=Acos(αx)+Bsin(αx) satisfies the endpoint conditions if and only if B=0 and α is a positive root of the equation tan(z)=1/z. These roots (a_{n})^{∞}_{1} are the abscissas of the points of intersection of the curves y=tan(x) and...

Homework Statement Consider a Hilbert space with a (not necessarily orthogonal) basis \{f_i\} Show that G=\sum_i |f_i\rangle\langle f_i| has strictly positive eigenvalues. Homework Equations The Attempt at a Solution I know that G=\sum_i |f_i\rangle\langle f_i| is hermitian...

Hello guise. I am familiar to a method of diagonalizing an nxn-matrix which fulfills the following condition: the sum of the dimensions of the eigenspaces is equal to n. As to the algorithm itself, it says: 1. Find the characteristic polynomial. 2. Find the roots of the characteristic...

Hello, In GR where does complex eigen values shows up its' place. I mean to say what is the practical usage of complex eigenvalues in General relativity. Also if anybody can explain where in GR, complex vector spaces come into picture. -- Shounak

Homework Statement Consider the interaction of two species of animals in a habitat. We are told that the change of the populations and can be modeled by the equations \frac{dx}{dt}= 0.1x-0.8y \frac{dy}{dt}=-0.2x+0.7y Find the solution to the above equations with initial values x(0)=9 and...

-2x + 3y + z = 0 3x + 4y -5z = 0 x -2y + z = -4 Find the characteristic equation, eigenvalues / eigenvectors of the system. I'm given to understand the eigenvalue problem is Ax = (lamba)x, but lamba doesn't exist in the system above. How can I solve for the eigenvalues when there are none?

Homework Statement Let v be a non-zero (column) vector in Rn. (a) Find an explicit formula for the matrix Pv corresponding to the projection of Rn to the orthogonal complement of the one-dimensional subspace spanned by v. (b) What are the eigenvalues and eigenvectors of Pv? Compute the...

Homework Statement I need to prove or disprove this statement: "If all eigenvalues of a diagonalizable matrix A are equal and have the same value c, then A=cI."Homework Equations The Attempt at a Solution I have tried coming up with a diagonalizable matrix that has eigenvalues with all the...

The eigenvalues are found by $$ \tan\lambda_n = \frac{1}{\lambda_n} $$ For large eigenvalues, the intersection get closer and closer to $\lambda_n = \pi k$ where $k\in\mathbb{Z}^+$ and $k > 15$. Is this correct? Without arbitrary picking a $k$, is there a better way to determine a $k$ for when...

Homework Statement I need to find the eigenvalues of the pion triplet under G parity Homework Equations G\mid\psi\rangle = CR_2\mid\psi\rangle The Attempt at a Solution OK so visually this problem is pretty simple, rotation about the 2 axis takes a pi+ to a pi- and then charge...

Homework Statement Find the principal stresses and the orientation for the principal axis of stress for the following cases of plane stress. σx = 4,000 psi σy = 0 psi τxy = 8,000 psi Homework Equations See picture. The Attempt at a Solution...

Hi! I'm reading David Tong's notes on QFT and I'm now reading on the chapter on the dirac equation http://www.damtp.cam.ac.uk/user/tong/qft/four.pdf and I stumbled across a statement where he claims that (\gamma^0)^2 = 1 \ \ \Rightarrow \text{real eigenvalues} while (\gamma^i)^2 = -1 \...

Homework Statement A=[0 -9; 1 -6] Can this matrix be diagonalized? Homework Equations det(A-\lambdaI)=0 The Attempt at a Solution det(A-\lambdaI)=0 gives the eigenvalues of the matrix and yields two eigenvalues that are equal, \lambda= -3 A matrix with repeating eigenvalues are...

This has already been adressed here: https://www.physicsforums.com/showthread.php?t=173896 , but I still didn't get the answer. The Harmonic Oscillator is fully described (according to my favourite QM book) by the HO Hamiltonian, and the commutation relations between the position and momentum...

Homework Statement Hello! I don't know how to solve this problem: find eigenvalues and eigenfunctions of quadratic membrane which is fixed in three edges. Fourth edge is flexible bended in the middle (at this edge membrane is in the shape of triangular). Surface tension of membrane is γ...

Homework Statement Given the matrix A: 4 2 2 2 4 2 2 2 4 Find the matrix P such that P-1AP is diagonal Homework Equations The Attempt at a Solution So I had this question today on a placement exam and it threw me for a loop. I found the eigenvalues to be 2,2, and 8. The...

Homework Statement If \lambda_i are the eigenvalues of a matrix A^2, and A is symmetric, then what can you say about the eigenvalues of A? Homework Equations The Attempt at a Solution I know how to prove that if \sqrt(\lambda_i) is an eigenvalue of A, then \lambda_i is an eigenvalue of A^2...

Assume P is a symmetric positive-definite matrix, and S to be a diagonal matrix with all its diagonal elements being greater than 1. Let Q = SPS then is Q-P symmetric positive-definite ? i.e. are the eigen-values of Q greater than P element-wise? or eig(Q)>= eig(P) in non-negative...

Homework Statement the order of eigenvalues is important, but when you calculate an eigenvalue polynomial i am still not aware of any rule that dictates which eigenvalue comes first and which does not. let me explain what i mean. take the matrix \begin{bmatrix} 3 & -2 \\ 5 & -4...

Let it be the X coordinate Pauli's matrix: \begin{array}{ccc} 0 & 1 \\ 1 & 0 \end{array} According to my calculations, it's eigenvectors would require that the spinor components to take the same value, but then, in order to have two orthogonal eigenvectors, we would need the complex...

Homework Statement A linear operator given (a matrix). That could be an orthogonal protection (that goes through the origin) or a symmetry with respect to a plane (that goes through the origin). 1-Get the eigenvalues of linear operator 2-Get the eigenspace associated with each eigenvalue...

Suppose an nxn matrix has n distinct eigenvectors vi when treated as a linear operator over ℝn. What is the relationship between these and the eigenvectors of the matrix when treated as a linear operator over ℝnxn, the space of nxn matrices? Since a matrix L acting on one with columns a1, a2...

Homework Statement Find the eigenvalues of: |13 -30 0| |1 0 0| |0 1 0| Homework Equations Equation for the eigenvalues: det(A-λI)=0 Cofactor expansion = det A = a11(a22a33-a23a32)+a12(a23a31-a21a33)+a13(a21a32-a22a31) The Attempt at a Solution |13-λ -30...

Homework Statement In each case describe the eigenvalues of the linear operator and a base in R^3 that consist of eigenvectors of the given linear operator. Write the matrix of the operator with respect to the given base. The Orthogonal Projection on the plane 2x + y = 0 and...

saravananbs's question from Math Help Forum, Hi saravananbs, No. The converse is not true in general. Take the two matrices, \(A=\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\mbox{ and }B=\begin{pmatrix}0 & 1\\1 & 0\end{pmatrix}\). \[\begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\begin{pmatrix}1 \\...

I'm looking into the stability of a system of ODEs, for which we've mannaged to extract a Jacobian matrix. Two of our eigenvalues are within our nummerical error tolerance, but they are close to zero. One of them is positive, which poses a problem for our stability analysis. We do know that...

Homework Statement Find the eigenvalues and the eigenfunctions for x^2y"+2xy'+λy = 0 y(1) = 0, y(e^2) = 0 Homework Equations See problem The Attempt at a Solution My book has one paragraph on this that does not help me. I tried using an auxiliary equation and solving for lambda...

I use the command eig(A) to calculate the eigenvalues of the matrix A, in which k,α,β,γ are variables, A=[-0.5*k -i*α 0;-i*α -0.5*γ -i*β;0 i*β -0.5k]; eig(A) but the MATLAB shows "Undefined function or variable 'k'." I don't know how to define the variables k,α,β,γ. Can anyone tell me...