Eigenvalues Definition and 853 Threads

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...

Homework Statement y^{(4)}+\lambda y=0 y(0)=y'(0)=0 y(L)=y'(L)=0 Homework Equations The hint says... let \lambda = -\mu ^4, \mu >0 or \lambda = 0The Attempt at a Solution Listening to the hint, I got r=\pm\mu With multiplicity 2 of each. So that means.. y=c_1 e^{\mu t}+c_2te^{\mu...

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...

Homework Statement http://img820.imageshack.us/img820/4874/cah.th.png Uploaded with ImageShack.us The Attempt at a Solution a) Did it already, 3 is the eigenvalue b) This is just finding the nullspace and the basis of the nullspace are my eigenvectors right? c) ignore...

given that 2 matrices have the same eigenvalues is it necessary that they be similar? If not, what is the connection between those 2?

Suppose I get the eigenvalues of A, which are \lambda_{1},\lambda_{2},\dots \lambda_{n}. Also, given any polynomial f(x), I get the eigenvalues of f(A). I'm trying to show that the eigenvalues of f(A) are f(\lambda_{1}),f(\lambda_{2}),\dots f(\lambda_{n}). Is this possible? How would I go about...

I am trying to relate eigenvalues with singular values. In particular, I'm trying to show that for any eigenvalue of A, it is within range of the singular values of A. In other words, smallestSingularValue(A) <= |anyEigenValue(A)| <= largestSingularValue(A). I've tried using Schur...

1.\frac{dx}{dt}= \stackrel{9 -12}{2 -1} x(0)=\stackrel{-13}{-5} So I seem to be having issues with this problem There are 2 eigenvalues that I obtained from setting Det[A-rI]=0 That gave me r^{2}-8r+15=0 solving for r and finding the roots i got (r-3)*(r-5)=0 so the...

Given a bounded domain with the homogeneous Neumann boundary condition, show that the Laplacian has an eigenvalue equal to zero (show that there is a nonzero function u such that ∆u = 0, with the homogeneous Neumann B.C.). I said: ∇•(u∇u)=u∆u+∇u2, since ∆u = 0, we have ∇•(u∇u)=∇u2 ∫...

Hello. I would like to numerically determine eigenvalues of a rectangular membrane which is twisted for \frac{\pi}{2}. Example picture: I'm solving Helmholtz equation: \nabla^2u+k^2u=0 where u=u(x,y) and \nabla^2 u=\frac{\partial^2u}{\partial x^2}+\frac{\partial^2v}{\partial y^2}...

Homework Statement using X''(x)+ lambda*X(x)=0 find the eigenvalues and eigenfunctions accordingly. Use the case lambda=0, lambda=-k2, lambda=k2 where k>0 Homework Equations X(0)=0, X'(1)+X(1)=0 The Attempt at a Solution I know that for lambda=0 X(x)=C1x+C2 which applying the...

Hi all, I am trying to find numerically the eigenvalues of a nonlinear schroedinger equation in a similar way as the Self Consistent Field method for Hatree-Fock problems. Does anybody know in the SCF calculation how to improve the convergency? Is there any trick other than simply inserting...

Homework Statement y'' + (lambda)y = 0, y'(0) = 0, y(1) = 0 We are told that all eigenvalues are nonnegative. Even with looking at the solution manual, I am unsure how to start setting these up. I've been starting by doing the following: y(x) = A cos cx + B sin dx y'(x) = -Ac sin(cx) + Bd...

Homework Statement See figure attached Homework Equations The Attempt at a Solution \lambda > 1, y^{''} + 2y^{'} + \alpha^{2}y = 0, \quad \alpha > 0 Into auxillary equation, m^{2} + 2m + \alpha^{2} = 0 I'm stuck as to how to solve this auxillary equation. Any...

Homework Statement Let F be a finite field of characteristic p. As such, it is a finite dimensional vector space over Z_p. (a) Prove that the Frobenius morphism T : F -> F, T(a) = a^p is a linear map over Z_p. (b) Prove that the geometric multiplicity of 1 as an eigenvalue of T is 1. (c) Let F...

So there's a circular helix parametrized by \vec x(t)=(a\cos(\alpha t), a\sin(\alpha t), bt) and you have the matrix K given in the Frenet-Serret formulas. In the book I'm reading it says that -\alpha^2 is the nonzero eigenvalue of K^2. Can someone explain how they know this is? I understand...

I need to know if there is any relationship between the positive definite matrices and its eigenvalues Also i would appreciate it if some one would also include the relationship between the negative definite matrices and their eigenvalues Also can some also menthow the Gaussian...

Consider two matrices: 1) A is a n-by-n Hermitian matrix with real eigenvalues a_1, a_2, ..., a_n; 2) B is a n-by-n diagonal matrix with real eigenvalues b_1, b_2, ..., b_n. If we form a new matrix C = A + B, can we say anything about the eigenvalues of C (c_1, ..., c_n) from the...

The usual eigenvalues of a LTV system does not say much about the stability but my intuition tells me there should be some kind of extension that applies to LTV systems as well. Like including some kind of inner derivative of the eigenvalues or something, I don't know... I guess in some way...

Homework Statement Matrix A has eigenvalues \lambda1= 2 with corresponding eigenvector v1= (1, 3) and \lambda2= 1 with corresponding eigenvector v2= (2, 7), find A. Homework Equations Definition of eigenvector: Avn=\lambdanvn The Attempt at a Solution I tried this by making...

Homework Statement Let V be the linear space of all real polynomials p(x) of degree < n. If p \in V, define q = T(p) to mean that q(t) = p(t + 1) for all real t. Prove that T has only the eigenvalue 1. What are the eigenfunctions belonging to this eigenvalue? Homework Equations Not sure...

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

Homework Statement Consider the Hamiltonian \hat{}H = \hat{}p2/2m + (1/2)mω2\hat{}x2 + F\hat{}x where F is a constant. Find the possible eigenvalues for H. Can you give a physical interpretation for this Hamiltonian? Homework Equations The Attempt at a Solution I don't think...

Homework Statement Let V be the vector space of all real-coefficient polynomials with degree strictly less than five. Find the eigenvalues and their geometric multiplicities for the following maps from V to V: a) G(f) = xD(f), where f is an element of V and D is the differentiation map...

Homework Statement http://i1225.photobucket.com/albums/ee382/jon_jon_19/Eigen.jpg The Attempt at a Solution It is a bit too long to type it all out, but I was wondering whether I am correct: I got, A = 7/2 , B = 0 , C = -1/8 , D = 1/8 And from this I worked out, x2(1) =...

Homework Statement dx/dt= -4x -y dy/dt= x-2y x(0)=4 y(0)=1 x(t)=? y(t)=? Homework Equations The Attempt at a Solution 1) find eigenvalues (x+4)(X+2)+1 X=-3,-3 2)eigenvectors: (-3-A)(x,y)=(0,0) eignvector=(-1,1) 3)using the P from this page...

From my Linear Algebra course I learned tha and eigenvalue w is an eigenvalue if it is a sollution to the system: Ax=wx, where A= square matrix, w= eigenvalue, x= eigenvector. We solved the system by setting det(A-I*w)=0, I=identity matrix Now in an advanced course I have come upon the...

Find the eigenvalues and corresponding eigenvectors of the following matrix. 1,1 1,1 Here is my attempt to find eigenvalues: 1-lambda 1 1 1-lambda Giving me: (Lambda)^2 -2(lambda) = 0 lambda = 0 lambda = 2 Is this correct??

Homework Statement Consider a particle with periodic boundary conditions of length L. Write dwon the expression for the normalised basis wave functions and their eigenvalues. Find the eigen value of the momentum and the expectation value of the momentum with respect to one of the momentum...

Homework Statement Show that the eigenvalues of a hermitian operator are real. Show the expectation value of the hamiltonian is real. Homework Equations The Attempt at a Solution How do i approach this question? I can show that the operator is hermitian by showing that Tmn =...

Homework Statement Let A \in \mathbb{C}^{n \times n} and set \rho = \max_{1 \le i \le n}|\lambda_i|, where \lambda_i \, (i = 1, 2, \dots, n) are the eigenvalues of A. Show that for any \varepsilon > 0 there exists a nonsingular X \in \mathbb{C}^{n \times n} such that \|X^{-1}AX\|_2 \le...

Homework Statement Hi there. I must give the eigenvalues and the eigenvectors for the matrix transformation of the orthogonal projection over the plane XY on R^3 So, at first I thought it should be the eigenvalue 1, and the eigenvectors (1,0,0) and (0,1,0), because they don't change. But I...

Homework Statement I am part way done with this problem... I don't know how to solve part e or part f. Any help or clues would be greatly appreciated. I have been trying to figure this out for a couple days now. W={<x,y,z>, x+y+z=0} is a plane and T is the orthogonal projection on it. a)...

Find a matrix that has eigenvalues 0,18,-18 with corresponding eigenvectors (0,1,-1), (1,-1,1), (0,1,1). ... I know the diagonlize rule, and the the rule to find a a power of A A= PDP^-1 D=P^-1AP ... but i am lost as to how to contine... help please?

Homework Statement Find the eigen values, eigenspaces of the following matrix and also determine a basis for each eigen space for A = [1, 2; 3, 4]Homework Equations \det(\mathbf{A} - \lambda\mathbf{I}) = 0 The Attempt at a Solution OK, so I found the eigenvalues and eigenspaces just fine...