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.
Suppose P: V->V s.t. P^2 = P and V = kerP + ImP (actually not just + but a direct sum). Find all eigenvalues of P.
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Which of the following explanations is right? (1 is an eigenvalue, but is 0 also?) Could somebody please explain?
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First answer:
Suppose that λ is an...
Homework Statement
Let A =
a b
c d
A characteristic value of A (often called an eigenvalue) is denoted by λ and satisfies the relation
det(A - λI) = 0
Obtain the characteristics values of E =
1 -1
-1 1
Homework Equations
Well I is the unit or identity matrix
1 0
0 1...
Homework Statement
Solve the system.
dx/dt=[1 -4; 4 -7]*x with x(0)=[3; 2]Homework Equations
The Attempt at a Solution
I am apparently not getting this at all. Can someone walk me through it? I konw I have to first find the eigenvalues and eigenvectors:
(1-λ)(-7-λ)+16=0
λ2+6λ+9=0
λ=-3,-3
So...
Homework Statement
solve the system dx/dt = [12 -6; 6 -3] with the initial value x(0) = [12; 9]
Homework Equations
The Attempt at a Solution
I know I need to find the Eigenvalues but then I get a little confused from there.
(λ-3)(λ+3)=0
λ=3, -3
Homework Statement
Given matrix A:
a 1 1 ... 1
1 a 1 ... 1
1 1 a ... 1
.. . .. ... 1
1 1 1 ... a
Show there is an eigenvalue of A whose geometric multiplicity is n-1. Express its value in terms of a.
Homework Equations
general eigenvalue/vector equations
The Attempt at a Solution
My...
Hi guys,
probably that's the wrong forum, but I was just curious about
the plot (Figure 1 Chapter XI A./1. page 1097 / Volume II) of the eigenvalues
E(\lambda).
If I calculate them they are supposed to be straight lines with positive or
negative slope i.e.:
E(\lambda) = E_n^0 + \lambda...
I'm trying to teach myself quantum mechanics using a book I got. I made an attempt at one of the questions but there are no solutions or worked examples so I'm wondering if I got it right.
Here it goes
Homework Statement
Suppose an observable quantity corresponds to the operator \hat{B}=...
I am somewhat confused about this property of an eigenvalue when A is a symmetric matrix, I will state it exactly as it was presented to me.
"Properties of the eigenvalue when A is symmetric.
If an eigenvalue \lambda has multiplicity k, there will be k (repeated k times),
orthogonal...
Homework Statement
Prove that similar square matricies have the same eigenvalues with the same algebraic multiplicities.
Homework Equations
C^-1PC=Q
The Attempt at a Solution
Am I supposed to show that (P-\lambdaI)x=(C^-1PC-\lambdaI)x?
Homework Statement
Let A be an nxn matrix and let I be the nxn identity matrix. Compare the eigenvectors and eigenvalues of A with those of A+rI for a scalar r.
Homework Equations
The Attempt at a Solution
I think I should be doing something like this:
det(A-\lambdaI), and...
I want to write myself a algorithm to solve generalised eigenvalue problems in quantum mechanics.I know there are a lot of library there that allow me to use it directly but i just want to write my own so that i can learn the mathematics methods that solve the problem...
I don't know how to...
Homework Statement
Prove that if two linear operators A and B commute and have non-degenerate eigenvalues then the two operators have common eigenfunctions.
Homework Equations
[A,B]= AB - BA= 0
Af=af
Bg=cg,\ let\ g=(f+1) --> B(f+1)=c(f+1)\ where\ a\neq c
The Attempt at a...
Homework Statement
Show that A and AT share the same eigenvalue.
Homework Equations
The Attempt at a Solution
let v be the eigenvector
Av=Icv
since ATv=ITcv
and IT=I,
ATv=Icv
so ATv=Icv=Av
so A and AT must have the same eigenvalue.
Homework Statement
Let U be a fixed nxn matrix and consider the operator T: Msub(n,n)------>Msub(n,n)
given by T(A)=UA.
Show that c is an eigenvalue of T if and only if it is an eigenvalue of U.
Homework Equations
The Attempt at a Solution
If T(A)=UA then T(A)-UA=0 (T-U)A=0.
Let...
Another proof...
Homework Statement
Suppose c is an eigenvalue of a square matrix A with eigenvector X=/=0.
Show that p(c) is an eigenvalue of p(A) for any nonzero polynomial p(x).
Homework Equations
The Attempt at a Solution
Knowing that c is an eigenvalue of A, it is true that...
Hi,
Is there any solution for the following problem:
Ax = \lambda x + b
Here x seems to be an eigenvector of A but with an extra translation vector b.
I cannot say whether b is parallel to x (b = cx).
Thank you in advance for your help...
Birkan
Hello,
I was reading something in my text/wikipedia, and they both said that "...the eigenvalues of a matrix are the zeros of its characteristic polynomial." Do they mean that λ in the characteristic polynomial causes det (A - λI) = 0 (in particular A = λI)?
JL
Theorem: Let A be in M_n_x_n(F). Then a scalar \lambda is an eigenvalue of A if and only if det(A - \lambda I_n) = 0.
Proof: A scalar lambda is an eigenvalue of A if and only if there exists a nonzero vector v in F^n such that lambda*v, that is (A - \lambda I_n)(v) = 0. By theorem 2.5, this...
Homework Statement
I would like to know what the definition of a Differential Eigenvalue Problem is please?
I am a maths undergraduate.
Homework Equations
\lambda y = L y, where \lambda is eigenvalue, L is a linear operator.
The Attempt at a Solution
I have searched via google...
Homework Statement
Let x be a unit vector. Namely x(Transpose)*x = 1. If (A − Let x be a unit vector. If (A − λI)x = b, then λ is an eigenvalue of A − bx(transpose).
The Attempt at a Solution
I have no idea where to start this proof.
Homework Statement
Let λ be an eigenvalue of A. Then λ^2 is an eigenvalue of A^2
The Attempt at a Solution
I know I have to start by using the fact that λ is an e.v of A then set up an equation relating the eigenvalues and vectors to A which is: Ax=λx. And I understand that the...
Homework Statement
"Let A be a diagonalizable n by n matrix. Show that if the multiplicity of an eigenvalue lambda is n, then A = lambda i"
Homework Equations
The Attempt at a Solution
I had no idea where to start.
Homework Statement
Let λ be an eigenvalue of A. Then λ+σ is an eigenvalue of A+σI
Homework Equations
The Attempt at a Solution
I'm guessing I need to use the fact that λ is an e.v of A to start with. But then when I add σ to both sides somehow I feel like I'm begging the question..
Homework Statement
If λ is and eigenvalue of the the matrix A then 3λ is an eigenvalue of 3A
Homework Equations
The Attempt at a Solution
. .
. λ is an e.v of A
Therefore, ∃ x not equal to 0 s.t Ax=λx
Then, 3Ax=3λx
which can written as 3(Ax)=3(λx)=λ(3x)
and 3x does not...
Homework Statement
Find a 3*3 matrix A which is not diagonalizable and such that 2 is the only eigenvalue of A
Homework Equations
The Attempt at a Solution
since λ=2,and it is a 3*3 matrix
i get the det(λI-A)=(λ-2)^3=0
then λ^3-6λ^2+12λ-8=0
now i use...
The problem is
A particle of mass m and electric charges q can move only in one dimension and is subject to a harmonic force and a homogeneous electrostatic field. The Hamiltonian operator for the system is
H= p2/2m +mw2/2*x2 - qεx
a. solve the energy eigenvalue problem
b. if the...
Homework Statement
This isn't a homework problem, just something I've been trying to conceptualize for a while. Can anyone exemplify with a physical analog the concept of eigenstates? For example, I know that eigenvalues of variables with continuous spectra do not exist in the physical...
Homework Statement
In Uniform Acceleration Motion, the force F is constant.
then potential V(x)=Fx, and Hamiltonian H=(p^2/2m)-Fx
The problem is to solve the eigenvalue problem Hpsi(x)=Epsi(x)
Homework Equations
F=constant
V(x)=Fx
H=(p^2/2m)-Fx
The Attempt at a Solution
I have...
Homework Statement
Let A be a 2x2 matrix for which there is a constant k such that the sum of the entries in each row and each column is k. Which of the following must be an eigenvector of A?
a. [1,0]
b. [0,1]
c. [1,1]
(The answer can be any or all of these)
The Attempt at a Solution
I...
Homework Statement
For the matrix A =
-1, 5
-2, -3
I found the eigenvalues to be -2 + 3i and -2 - 3i.
Now I need some help to find the eigenvectors corresponding to each.
Homework Equations
The Attempt at a Solution
For r = -2 + 3i, I plugged that into the (A - Ir) matrix...
Hello everybody,
I have a question for which I cannot find the answer around,
any help would be really appreciated.
Suppose we have a matrix A of a linear transformation of a vector space,
with only one eigenvalue, say 's'.
My question is: Is the operator (A-sI) nilpotent? ('I' is the...
To give you some background, I am trying to perform an AHP calculation using Java code. I have a 15x15 matrix and I need to find its eigenvector. I want the eigenvector that corresponds to the greatest eigenvalue.
Let's say I already have some method that gives me all the eigenvectors and all...
The wave function of "particle in a box" is Asin(kx).
Since potential energy is zero inside the box, so the Hamiltonian is just kinetic energy
In principle, I should be able to find eigenvalue of momentum using momentum operator,
but stymied in solving the equation. Can somebody help me find...
Hi everyone, I am stuck with the following for last couple of days.
Many books mention during the development in the idea of Eigenvalue problem: say, you have the equation
[\ A-\lambda\ I]\ X=\ 0 where A is an NxN matrix and X is an Nx1 vector.
The above consists of n equations.Say,all...
Homework Statement
3.) Stress analysis at a critical point in a machine member gives the three-dimensional state of stress in MPa as the following:
y =
[ 105 0 0
0 -140 210
0 210 350 ]...
Hello,
I am using COMSOL (RF modul) for some time now to calculate the eigenvalues (modes) of optical fibers. So far I changed all the params from hand in the software.
But it is getting anoying to change a parameter (the wavelength) and to recalculate if one wants to calculate for many...
Homework Statement
Bessels equation of order n is given as the following:
y'' + \frac{1}{x}y' + (1 - \frac{n^2}{x^2})y = 0
In a previous question I proved that Bessels equation of order n=0 has the following property:
J_0'(x) = -J_1(x)
Where J(x) are Bessel functions of...
Homework Statement
Prove that an orthogonal transformation T in Rm has 1 as an eigenvalue if the determinant of T equals 1 and m is odd. What can you say if m is even?
The attempt at a solution
I know that I can write Rm as the direct sum of irreducible invariant subspaces W1, W2, ..., Ws...
Hi
I am supposed to, without calculation, find 2 linearly independent eigenvectors and a eigenvalue of the following matrix A
5 5 5
5 5 5
5 5 5
The eigenvalue is easy -- it is 15. And I can find one eigenvector, [1 1 1] (written vertically), but another without calculation? Is there...
Homework Statement
solve the eigenvalue problem
∫(-∞)x dx' (ψ(x' ) x' )=λψ(x)
what values of the eigenvalue λ lead to square-integrable eigenfunctions?
The Attempt at a Solution
∫(-∞)xdx' (ψ(x' ) x' )=λψ(x)
differentiate both sides to get
ψ(x)x=λ d/dx ψ(x)
ψ(x)x/λ=...
Homework Statement
Find all eigenfunction of momentum operator in x(px=h/i*d/dx) and their eigenvalues.
Homework Equations
operator*eigenfunction=eigenvalue*eigenfunction
Operator=px
The Attempt at a Solution
I really don't have any clues
Thank you
I read from a book and claim that for any hermitian matrix can be diagonalized by a unitary matrix whose columns represent a complete set of its normalized eigenvectors. It then given an equation...
Homework Statement
This is the original question:
\frac{d^{2}y}{dx^{2}}-\frac{6x}{3x^{2}+1}\frac{dy}{dx}+\lambda(3x^{2}+1)^{2}y=0
(Hint: Let t=x^{3}+x)
y(0)=0
y(\pi)=02. The attempt at a solution
This might be all wrong, but this is all I can think of
\frac{dt}{dx}=3x^{2}+1
so...
Solve the eigenvalue problem
\frac{d^2 \phi}{dx^2} = -\lambda \phi
subject to
\phi(0) = \phi(2\pi)
and
\frac{d \phi}{dx} (0) = \frac{d \phi}{dx} (2 \pi).
I had the solution already, but am looking for a much simpler way, if any.
EDIT:
Sorry that I accidentally posted...
If a vector v\in V and a linear mapping T:V\to V are fixed, and there exists numbers \lambda_1\in\mathbb{C}, n_1\in\mathbb{N} so that
(T - \lambda_1)^{n_1}v = 0,
is it possible that there exists some \lambda_2\neq\lambda_1, and n_2\in\mathbb{N} so that
(T - \lambda_2)^{n_2}v = 0?
(Here...
Homework Statement
V is an eigenvector of the nxn matrix A, with a eigenvalue of 4. explain why V is a eigenvector of A^2+2A+3I. what is the associated eigenvalue?
Homework Equations
The Attempt at a Solution
is the eigenvalue of A^2+2A+3I=21?
Homework Statement
Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace.
T is the transformation on R2 that reflects points across some line through the origin.
The Attempt at a Solution
Since they tell us that...