What is Eigenvalue: Definition and 400 Discussions
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by
λ
{\displaystyle \lambda }
, is the factor by which the eigenvector is scaled.
Geometrically, an eigenvector, corresponding to a real nonzero eigenvalue, points in a direction in which it is stretched by the transformation and the eigenvalue is the factor by which it is stretched. If the eigenvalue is negative, the direction is reversed. Loosely speaking, in a multidimensional vector space, the eigenvector is not rotated.
Homework Statement
Suppose B is a real 2x2 matrix with the following eigenvalue:
\frac{√3}{2} + \frac{3i}{2}.
Find B^3.
Homework Equations
One of the hints is to consider diagonalization over C together with the fact
that (\frac{1}{2} + \frac{√3}{2}i)^3 = -1.
The Attempt...
Homework Statement
Write down the v=1 eigenfunction for the harmonic oscillator. Substitute this eigenfunction into the Schrodinger equation and show that the eigenvalue is (3/2)hν.
Homework Equations
The Attempt at a Solution
I'm not really sure on how to to this, but here's...
Homework Statement
Show that the following is an eigenfunction of \hat{H}_{QHO} and hence find the corresponding eigenvalue:
u(q)=A (1-2q^2) e^\frac{-q^2} {2}
Homework Equations
Hamiltonian for 1D QHO of mass m
\hat{H}_{QHO} = \frac{\hat{p}^2}{2m} + \frac{1}{2} m \omega^2 x^2...
I know that the eigenvalue of a linear system is a scalar such that Ax=λx. I know many ways to find the eigenvalue of a linear system. But I'm pulling my hair out trying to figure out what it is actually telling me about the system.
Can anyone give me a non-technical straight up answer on why...
Hey,
My question is on the probability of attaining a particular eigenvalue for the total angular momentum operator squared for a particular state ψ, the question is shown in the image below:
I believe the eigenvalue of the total angular momentum operator squared is given by j(j+1)...
Homework Statement
Here is the question along with my work. I attempted to solve for the actual solution using both eigenvectors. From what I have been taught it should yield the same answer... But as you can see (circled in red) the solutions are clearly different. Is this normal or maybe...
Use finite difference method to solve for eigenvalue E from the following second order ODE:
- y'' + (x2/4) y = E y
I discretize the equation so that it becomes
yi-1 - [2 + h2(x2i/4)] yi + yi+1 = - E h2 yi
where xi = i*h, and h is the distance between any two adjacent mesh points.
This...
Use finite difference method to solve for eigenvalue E from the following second order ODE:
- y'' + (x2/4) y = E y
I discretize the equation so that it becomes
yi-1 - [2 + h2(x2i/4)] yi + yi+1 = - E h2 yi
where xi = i*h, and h is the distance between any two adjacent mesh points.
This is my...
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...
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
We are to show that for 0<β<1, eigenvalues are strictly positive and for β>1, we have to determine how many negative eigenvalues there are.
u''+λ2u=0, u(0)=0, βu(π)-u'(π) = 0
Homework Equations
I've already shown that the eigenvalues are determined by tan(λπ)=λ/β (was told...
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
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...