What is Eigen vectors: Definition and 30 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.
This refers to problems A.28/29 from Quantum Mechanics – by Griffiths & Schroeter.
I’ve now almost finished the Appendix of this book and been greatly helped with the problems by Wolfram Alpha.
In problems A.28/29 we are asked to "Construct the unitary matrix S that diagonalizes T" where T is...
Solved (sorry i tried again and realized my E-values were wrong)
1. Homework Statement
Homework Equations
Find Eigen Values and then what?
The Attempt at a Solution
I got eigen values as 3 and -3.
Now how to proceed?
I got Eigen Vector as: 1, 1 for eigen value of 3
and eigen vector as 8, 2...
In non-relativistic QM, say we are given some observable M and some wave function Ψ. For each unique eigenvalue of M there is at least one corresponding eigenvector. Actually, there can be a multiple (subspace) eigenvectors corresponding to the one eigenvalue.
But if we are given a set of...
Hi, what is the physical meaning, or also the geometrical meaning of the inner product of two eigenvectors of a matrix?
I learned from the previous topics that a vectors space is NOT Hilbert space, however an inner product forms a Hilbert space if it is complete.
Can two eigenvectors which...
So I ran into an case I have not seen before. Say we have a system of 3 equations such that W´=AW, where W=(x(t),y(t),z(t)) and A is a 3x3 matrix. The way I usually approach these is by finding the eigenvalues of A to then find the eigenvectors and thus find the ¨homogenous¨ solution. What...
My professor states that "A matrix is diagonalizable if and only if the sum of the geometric multiplicities of the eigen values equals the size of the matrix". I have to prove this and proofs are my biggest weakness; but, I understand that geometric multiplicites means the dimensions of the...
Homework Statement
Find all values a\in\mathbb{R} such that vector space V=P_2(x) is the sum of eigenvectors of linear transformation L: V\rightarrow V defined as L(u)(x)=(4+x)u(0)+(x-2)u'(x)+(1+3x+ax^2)u''(x). P_2(x) is the space of polynomials of order 2.
Homework Equations
-Eigenvalues and...
Question
Consider the matrix $$
\left[
\matrix
{
0&0&-1+i \\
0&3&0 \\
-1-i&0&0
}
\right]
$$
(a) Find the eigenvalues and normalized eigenvectors of A. Denote the eigenvectors of A by |a1>, |a2>, |a3>. Any degenerate eigenvalues?
(b) Show that the eigenvectors |a1>, |a2>, |a3> form an...
What does it mean when it says eigenvalues of Matrix (3x3) A are the square roots of the eigenvalues of Matrix (3x3) B and the eigenvectors are the same for A and B?
I want to know what exactly Eigen value imply. What is its Physical significance ? Physical significance of eigen vector? Does eigen value concept apply in signal processing or evalvating frequency response off a system?
Homework Statement
Consider the Hamiltonian:
$$\hat{H}=C*(\vec{B} \cdot \vec{S})$$
where $C$ is a constant and the magnetic field is given by
$$\vec{B} = (0,B,0) $$
and the spin is
$$\vec{S} = (\hat{S}_{x},\hat{S}_{y},\hat{S}_{z}),$$
with$$\hat{S}_{x}...
Hi all,
I have a doubt regarding the physical significance of eigen vectors of the covariance matrix. I came to know that eigen vectors of covariance matrix are the principal components for dimensionality reduction etc, but how to prove it?
I am studying an article http://arxiv.org/abs/quant-ph/9907069 and having some problems understanding it.
Is self adjointness of an operator a sufficient or necessary and sufficient requirement for its eigen vectors with the generalized eigenvectors (i don't know what are these) to form...
Hello to all of you,
Is there a way to get the matrix A=[a b c d] from the eigenvectors (orthogonal) matrix
H= sin(x) cos(x)
cos(x) -sin(x)
or to pose it differently to find a matrix that has these 2 eigenvectors ?
Thank you in advance .
Michael
Normal modes of vibration, two masses, two spring, arranged vertically with m2 at the top, m1 underneath arranged (top to bottom) m2, k2, m1, k1, rigid support
I have solved the first part of an undamped coupled spring problem to give
m_1m_2 \omega ^ 4 + ((m_1+m_2)k_2+m_2k_1)\omega ^2...
Homework Statement
Find the rank off matrices?
i)A=[2 0 9 2; 1 4 6 0; 3 5 7 1 ] 3X4
ii)A=[3 1 4; 0 5 8; -3 4 4; 1 2 4;] 4X3
Find Eigen Vectors and Values of A;
A = [3 2 4; 2 0 2; 4 2 3 ]
Homework Equations
-when det(A) is not equal to zero it will the rank of matrices...
I have a matrix and can't seem to get my head around finding all the eigen vectors.
The matrix is A:
(1 0 0 0
1 0 0 1
0 1 0 0
0 0 1 0)
I got the eigen values as:
λ1 = 1, λ2 = λ3 = λ4 = 0
For λ1:
The eigen vector V1 is (0, 1, 1, 1).
For λ2 -> λ4:
The only eigen vector I...
Hello, this is related to my research on RCW(FMM) analysis of light shone onto crossed gratings.
main question: can you exploit having matrix A and its inverse A^{-1} to calculate the eigen vectors/values of A in a more effective way?
backgroud:
so my problem is this: I have an unusual...
Hi,
I have a square symmetrical matrix A (ugly I know)
321.1115, -57.5311, -33.9206
-57.5311, 296.7836, 10.8958
-33.9206, 10.8958, 382.1050
which has the eigen values,
248.8034
341.6551
409.5415
Am I right in saying that A...
For each of the following linear operators T on a vector space V and ordered bases beta, compute [T]beta, and determine whether beta is a basis conisting of eigen vectors of T.
V=R^2, T((a,b)^t)= (10a-6b
17a-10b)
and beta ={(1,2)^t , (2,3)^t)
im...
Let a state vector \psi is eigen vector for a Hamiltonian H which governs the Schrodinger equation (in its general form)of a system. Then, will probability distribution of \psi w.r.t any observable remain unchanged as time evolves?
Homework Statement
This is probably a really simple question , but how do i test my eigen values to see if there right ,---------- (A-tI)x=0 where t is an eigen value , I know how to test if my eigen vectors are correct but how do i test to see if my eigen values are right ...
Hi all,
I am a complex systems researcher and I need to have complete knowledge about eigen vectors and eigen values. How does change in dimension affect a point's eigen vector and eigen value? What does principal eigen vector and principal eigen value mean for a point of n-dimension...
Homework Statement
det(K-lambdaM)=0
K is stiffness and M is mass matrix
Homework Equations
How do I make sure that Q'*M*Q=I
Where Q is the matrix whose column entries are the eigen vectors.
The Attempt at a Solution
I want to know how to do it without MATLAB help. Is it always...
Homework Statement
(a) Find the characteristic equation and
(b) the eigenvalues and corresponding eigen-vectors
[-5 0 0
3 7 0
4 -2 3]
Homework Equations
det(\lambda*I - A)
The Attempt at a Solution
Finding the characteristic equation wasn't that challenging, I took...
In Quantum mechanics, we frequently deal with eigen value equations. When we speak of eigen value equations, we come across terms like eigen values,eigen vectors,eigen functions etc. When an operator is operated on certain quantities we get the same quantity multiplied by a constant. These...
Homework Statement
Question 1:
A) Show that if A is diagonalizable then A^{T} is also diagonalizable.
The Attempt at a Solution
We know that A is diagonalizable if it's similar to a diagonal matrix.
So
A=PDP^{-1}
A^{T}=(PDP^{-1})^{T}
which gives
A^{T}=(P^{-1})^{T}DP^{T} as...
Homework Statement
A mixing protocol for 5 containers of solvent consists of a repetition of the following procedure. The contents
of each container are removed and divided up into pre-defined fractions. The various fractions are then poured back
into the containers in a pre-assigned way...
I am trying to find eigen values and eigen vectors for A
Its 2X2 matrix. A first row (16 -10) second row (-10 24)
I got Eigen values as 30.77 and 9.22 but when i try to find eigen vectors here are the equations I end up with
-14.77v1 - 10v2= 0
-10v1 - 6.77v2 = 0
Kinda confused how to...