Eigenvalue Definition and 382 Threads

Homework Statement Calculate the eigenvalues of the matrix 5 2 -3 0 Homework Equations The Attempt at a Solution Ok we were taught that eigenvalues were calculated by taking the determinant( A - λI) = 0. So just subtract a "λ" value from the diagnol entries of the given...

Can someone help me with this question? I know we have to set up the auxiliary equation and then solve for λ but for some reason I am not getting the right answer. My equation is: m^2 + 4m + (5λ + 3) = 0 then we get -2 ± sqrt(5λ-1)i Now can somebody explain what I have to do...

Homework Statement Show that if a 6x6 matrix A has a negative determinant, then A has at least one positive eigenvalue. Hint: Sketch the graph for the characteristic polynomial of A. Homework Equations Characteristic polynomial: (-\lambda)^n + (\text{tr}A)(-\lambda)^{n-1} + ...

Show that a matrix and its transpose have the same eigenvalues. I must show that det(A-λI)=det(A^t-λI) Since det(A)=det(A^t) →det(A-λI)=det((A-λI)^t)=det(A^t-λI^t)=det(A^t-λI) Thus, A and A^t have the same eigenvalues. Is the above enough to prove that a matrix and its transpose have the...

If ##\hat{A}\vec{X}=\lambda\vec{X}## then ##\hat{A}^{-1}\vec{X}=\frac{1}{\lambda}\vec{X}## And what if ##\lambda=0##?

Suppose a square matrix A is given. Is it true that the null space of A corresponds to eigenvectors of A being associated with its zero eigenvalue? I'm a bit confused with the terms 'algebraic and geometric multiplicity' of eigenvalues related to the previous statement? How does this affect the...

Homework Statement Find the energy eigenvalue. Homework Equations H = (p^2)/2m + 1/2m(w^2)(x^2) + λ(x^2) Hψ=Eψ The Attempt at a Solution So this is what I got so far: ((-h/2m)(∂^2/∂x^2)+(m(w^2)/2 - λ)(x^2))ψ=Eψ I'm not sure if I should solve this using a differential...

Homework Statement Hey, guys. I'm having trouble finding the general solution to a second order, homogeneous ODE. It is the first step to solving an eigenvalue problem and my professor is about as much help as a hole in the head. I've tried multiple "guesses" and have combed various...

The probability of measuring a value a for an observable A if the system is in the normalized state |\psi\rangle is |\langle a|\psi\rangle|^2 where \langle a| is the normalized eigenbra with eigenvalue a. This is more-or-less the formulation of the Born rule as it appears in my text. But...

Homework Statement Let there be 3 vectors that span a space: { |a>, |b>, |c> } and let n be a complex number. If the operator A has the properties: A|a> = n|b> A|b> = 3|a> A|c> = (4i+7)|c> What is A in terms of a square matrix? Homework Equations det(A-Iλ)=0 The Attempt...

Hey guys, I've been trying to brush up on my linear algebra and ran into this bit of confusion. I just went through a proof that an operator with distinct eigenvalues forms a basis of linearly independent eigenvectors. But the proof relied on a one to one mapping of eigenvalues to...

For example, ODE: y'' + y = 0 solve this problem using MAPLE f(x) = _C1*sin(x)+_C2*cos(x) My question is Eigenvalue for D^2+1=0 is +i, -i so general solution is f(x) = C1*exp(i*x)+C2*exp(-i*x) according to Euler's formula f(x) = C1( cos(x)+i*sin(x) ) + C2*( cos(x)-i*sin(x) ) it is different...

I'd like to show that if there exists some operator \overset {\wedge}{x} which satisfies \overset {-}{x} = <\psi|\overset {\wedge}{x}|\psi> , \overset {\wedge}{x}|x> = x|x> be correct. \overset {-}{x} = \int <\psi|x> (\int<x|\overset {\wedge}{x}|x'><x'|\psi> dx')dx = \int <\psi|x>...

Homework Statement I have to find for which "a" an eigenvalue for the following system is 0. The system: 1 -1 1 -1 2 -2 0 a 1 Homework Equations My characterstic equation: (1-λ)(2-λ)(1-λ)+2a -(1-λ) -a = 0The Attempt at a Solution I then proceed: (1-λ)(λ2-3λ-2+a) = 0 but then I'm kind...

difference between eigenvalue and an expectation value of an observable. in what circumstances may they be the same? from what i understand, an expectation value is the average value of a repeated value, it might be the same as eigen value, when the system is a pure eigenstate.. am i right?

The solution to a linear differential equation is, y=exp(ax). If a is complex ,say a=b+ic, then the period is T=2pi/c. My question is, if a is in polar form, a=r*exp(iθ), how is the period then T=2pi/θ. Any help would be great, Thank, Will

Homework Statement We have a matrix Anxn (different than the identity matrix I) and a scalar λ=1. We want to check if λ is an eigenvalue of A. Homework Equations As we know, in order for λ to be an eigenvalue of A, there has to be a non-zero vector v, such that Av=λv The Attempt at a Solution...

Consider a generalized Eigenvalue problem Av = \lambda Bv where $A$ and $B$ are square matrices of the same dimension. It is known that $A$ is positive semidefinite, and that $B$ is diagonal with positive entries. It is clear that the generalized eigenvalues will be nonnegative. What else can...

Hi all, What is the normal procedure to verify that I got the correct results (eigenvalues and eigen vectors) from the eigenvalue problem? I'm using the lapack library to solve eigenvalue problem summarized below. I've 2 matrices K and M and I get the negative results for eigenvalues...

Homework Statement Use the power method to calculate the dominant eigenvalue and its corresponding eigenvectors for the matrices. The questions are attached with this thread. I have attempted both and seem to have done the first question correctly. I am attempting the second question and am...

Say I have a 4x4 matrix and I know 3 eigenvalues and the 3 corresponding eigenvectors. Is there a fast way to calculate which one has multiplicity 2 without calculating the characteristic polynomial(too time consuming for a 4x4 matrix) or without determining the dimensions of (A-λ I) for each...

Can I have a matrix that has an uncountable number of eigenvalues? If the matrix was infinite. And also can I have a matrix with a countable number of rows and an uncountable number of columns?

hi community, In general I would like to know the what's the general way of solving the moderate or large-scale eigenvalue or algorithms in structural dynmics. The simple motion equation is as follow. M*d2X(t)/dt2+C*dX(t)/dt+K*X(t) = F(t). The bolded expressions are known before the...

I have got a problem in my research. For the following matrix, a a a a a a b b b b a a a a a a b b b b a a a a a a b b b b a a a a a a b b b b a a a a a a a a a a a a a a a a a a a a b b b b a a a a a a b b b b a a a a a a b b b b a a a a a a b b b b a a a a a a, does anyone know how...

Hey everyone, I have a problem with over thinking things quite often, so I once again need help haha. How would you go about proving this: λ=0 is the only eigenvalue of A \Rightarrow Ax=0 \forallx Any help would be appreciated! Thanks

http://dl.dropbox.com/u/33103477/question.png I have determined the eigenvalues which are -2, 2 and 1 respectively. I'm pretty sure that the one with multiplicity of 2 is the, the eigenvalue = 2 as it occur's twice in the diagnol. But I don't think that's a concrete enough reason. Any...

Homework Statement I have a problem u'' + lambda u = 0 with BCs: u'(0) = b*u'(pi), u(0) = u(pi). where b is a constant. I have to find b which makes the BCs and problem self-adjoint. Homework Equations see below The Attempt at a Solution I see in my notes...

Homework Statement Find the eigenvalues and eigenvectors of the following matrix: M = 1 1 0 1 Can this matrix be diagonalised? Homework Equations The Attempt at a Solution The characteristic equation is (1 - \lambda)^{2} = 0 which gives \lambda = 1. Substitute \lambda = 1 and...

Homework Statement The 2x2 matrix representing a rotation of the xy-plane is T = cos θ -sin θ sin θ cos θ Show that (except for certain special angles - what are they?) this matrix has no real eigenvalues. (This reflects the geometrical fact that no vector in the plane is...

Homework Statement Equations: \frac{dv_{1}}{dt} = -v_{1} - \frac{2v_{2}}{3} + 1 + \frac{t}{3} \frac{dv_{2}}{dt} = -2v_{2} - 1 - 2t Initial Conditions: v_{1}(0) = 6 v_{2}(0) = -6 2. The attempt at a solution Defined the following: v(t) = [ v_{1}(t) v_{2}(t) ] \frac{dv(t)}{dt} = [...

Homework Statement Let T:P2→P2 be defined by T(a0+a1x+a2x2)=(2a0-a1+3a2)+(4a0-5a1)x + (a1+2a2)x2 1) Find the eigenvalues of T 2) Find the bases for the eigenspaces of T. I believe the 'a' values are constants. Homework Equations None. The Attempt at a Solution The problem I am...

Hi all, I need to find the λ and the ai that solves the Generalized eigenvalue problem [A]{a}=-λ2 [B]{a} with [A]= -1289.57,1204.12,92.5424,-7.09489,-25037.4,32022.5,-10004.3,3019.17 1157.46,-1077.94,-0.580522,-78.9482,32022.5,-57353.5,36280.6,-10949.6...

I intend to use the Gershgorin Circle Theorem for estimating the eigenvalues of a real symmetric (n x n) matrix. Unfortunately, I'm a bit confused with the examples one might find on the internet; What would be the mathematical formula for deriving estimates on eigenvalues? I understand that...

Hello, it's been a while since i did linear algebra. i need some help. I have this matrix: 1 1 0 0 1 0 0 0 0. I know the eigenvalues are 1,1,0; and that the eigenvectors will be: (1,0,0), (0,0,0) and (0,0,1). But I cannot do the jordan decomposition on the matrix i.e. write it in the form...

Hello I have this Hamiltonian: \mathcal{H} = \alpha S_{+} + \alpha^{*}S_{-} + \beta S_{z} with \alpha, \beta \in \mathbb{C} . The Operators S_{\pm} are ladder-operators on the spin space that has the dimension 2s+1 and S_{z} is the z-operator on spin space. Do you know how to get (if...

Homework Statement The Attempt at a Solution So, first I wrote, T(X) = λ_1 X, T(Y) = λ_2 Y If λ_1 = λ_2: T(X+Y) = T(X) + T(Y) = λ_1 X + λ_2 Y = λ_1 (X+Y), so this does indeed seem to be an eigenvector. But I'm less convinced for the case λ_1 ≠ λ_2. Again, I get the...

I posted a problem called "estimating eigenvalue of perturbed matrix" in the section 'Linear and abstract algebra' cus, well it was to do with matrices (I'm a physicist - appologies for that). Actually come to think of it the problem probably has more to do with analysis...If a kind maths peep...

Say M_{ij} = A_{ij} + s B_{ij}/2, where the matrices are 3 by3 and A_{ij} symmetric, s \in [0,s^*], and the smallest eigenvalue of A is lambda \leq -(1/2). Given that |M_{ij} - A_{ij}| \leq to C_{s^*} s/2 and |A_{ij}| \leq 1, plus that the cubic equation determining the eigenvalues has an...

Homework Statement You are given a self-adjoint operator \hat{A} and the equation and \hat{A}\Phi_{i} ~=~ \Phi_{i}a_{i}. Prove that ai are real numbers. Homework Equations There are instructions to guide me along with the question. The first step it says to do is write the eigenvalue...

So, my problem statement is: Suppose that two operators P and Q satisfy the commutation relation [Q,P] = Q . Suppose that ψ is an eigenfunction of the operator P with eigenvalue p. Show that Qψ is also an eigenfunction of P, and find its eigenvalue. This shouldn't be too difficult, but...

Homework Statement Show that the equation below is an eigenfunction for the Quantum Harmonic Oscillator Hamiltonian and find its corresponding eigenvalue. Homework Equations u1(q)=A*q*exp((-q^{2})/2) The Attempt at a Solution Ok, so I know that the Quantum Harmonic Oscillator...

Assuming the matrix is positive definite (necessary for cholesky decomposition). Which is faster? Which is more accurate? Is there a reliable source that has all the most common decompositions listed in order of accuracy and speed?

I've been trying to invert a real symmetric matrix and the inverse that I compute via eigenvalue decomposition is not the inverse (using QV^-1Q^T), the stranger thing is that QVQ^T gets back my orginal matrix matrix. Even more unusual is that the matrix starts off at approximately identity (in...

function: e^-(x^2/2) operator: d^2/dx^2 -x^2 The answer key says the function is an eigenfunction of the operator with an eigenvalue of -6. I can't figure out how to reach this conclusion. Also, Wolfram Alpha says d/dx(d/(dx)e^(-x^2/2)) = e^(-x^2/2) (x^2-1). Isn't this inconsistent with...

Homework Statement αo, α1,..., αd \inℝ. Show that αo + α1λ + α2λ2 + ... + αdλd \inℝ is an eigenvalue of αoI + α1A + α2A2 + ... + αdAd \inℝ^{nxn}. 2. The attempt at a solution If λ is an eigenvalue of A, then |A - Iλ| = 0. Also, λn is an eigenvalue An. So we basically have to somehow...

Proof involving nonsingular matrices. Homework Statement If (I + A) is nonsingular, prove that (I - A)(I + A)-1 = (I + A)-1(I - A), and hence (I - A)/(I + A) is defined for the matrix. I've proved it like this: Let (I - A)(I + A)-1 = A, and (I + A)-1(I - A) = B. B-1 = (I - A)-1(I +...

Hello, Given the hamiltonian : H = -( aS_z^2 + b(S_+^2 +S_-^2) ) with S=1 and a,b>0 are constants. working with the base: { |m=1> , |m=-1> , |m=0> } The matrix form of H is: H = \left( \begin{array}{ccc} -ah^2 & -bh^2 & 0 \\ -bh^2 & -ah^2 & 0 \\ 0 & 0 & 0 \end{array}...

hi i know how to calculate eigenvalue of given matrix. I want to know if two non homogenous simutaneous equation are given - than can we find its eigenvalue.

we are given B = CAC^-1 Prove that A and B have the same characteristic polynomial given a hint: explain why ƛIn = CƛInC^-1 what I did was: B = CAC^-1 BC = CA Det(BC) = Det(CA) Det(B) Det(C) = Det(C) Det(A) Now they’re just numbers so I divide both sides by Det(C) Det(B) = Det(A)...

Homework Statement Let Q be an orthogonal matrix with an eigenvalue λ_{1} = 1 and let x be an eigenvector belonging to λ_{1}. Show that x is also an eigenvector of Q^{T}. Homework Equations Qx = λx where x \neq 0 The Attempt at a Solution Qx_{1} = x_{1} for some vector x_{1}...