What is Eigenvalues: Definition and 850 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.
:rolleyes: :cool: I have a question..yesterday at Wikipedia i heard about the "Hermite Polynomials2 as Eigenfunctions of Fourier (complex?) transform with Eigenvalues i^{n} and i^{-n}...could someone explain what it refers with that?...when it says "Eigenfunctions-values" it refers to the...
I'm having trouble with this: Prove that if P is a linear map from V to V and satisfies P^2 = P, then trace P is a nonnegative integer.
I know if I find the eignevalues , their sum equals trace P. But how do I find them here?
any thoughts?
Thanks
Why do they have to purely imaginary?
I got a proof that looks like Ax=ax
where a = eigenvalue
therefore Ax.x = ax.x = a|x|^2
Ax.x = x.(A^t)x
where A^t = transpose = -A
x.(-A)x = -b|x|^2
therefore a=-b, where b = conjugate of a
Now is this as far as i need to go?
Please help me I am hopelessly lost and don't even know where to start! I guess they're right when they said girls suck at math! It's not fair! :redface:
Let A be an nxn matrix with only real eigenvalues. Prove that there is an orthogonal matrix Q such that (Q^T)AQ is upper triangular with...
matrix A = \left(\begin{array}{ccc}3&0&0 \\ 0&3&0 \\3&0&0 \end{array}\right)
has two real eigenvalues lambda_1=3 of multiplicity 2, and lambda_2=0 of multiplicity 1. find the eigenspace.
A = \left(\begin{array}{ccc}3-3 &0&0 \\ 0&3-3&0 \\3&0&0-3 \end{array}\right)
A =...
I need a bit of help with these boundary value problems. I'm trying to find their eigenvalues and eigenfunctions and although I pretty much know how to do it, I want to exactly WHY I'm doing each step. I attached part of my work, and on it I have a little question next to the steps I need...
If I have two positive definite Hermitian NxN matrices A and B, if I adiabatically change the components of A to B (constraining any intermediate matrices to be Hermitian as well, but not necessarily positive definite) while \"following\" the eigenvalues ... will the mapping of the eigenvalues...
"Suppose V is a (real or complex) inner product space, and that T:V\rightarrow V is self adjoint. Suppose that there is a vector v with ||v||=1, a scalar \lambda\in F and a real \epsilon >0 such that
||T(v)-\lambda v||<\epsilon.
Show that T has an eigenvalue \lambda ' such that |\lambda...
In my textbook recently I stumbled across the following:
Give a general description of those matrices which have two real eigenvalues equal in 'size' but opposite in sign? Could anyone explain this again very simply :-)
Hey!
Does someone know of some resources which describe how to code a function which calculates the eigenvalues of a matrix? This could be either resources on the net or a book. If you know of a good book which teaches about programming and mathematics together in general I'd be happy to know...
Show that every eigenvalue of A is zero iff A is nilpotent (A^k = 0 for k>=1)
i m having trouble with going from right to left (left to right i got)
we know that det A = product of the eignevalues = 0
when we solve for the eigenvalues and put hte characteristic polynomial = 0
then
det...
I'm asked to find the eigenvalues and eigenvectors of an nxn matrix. Up until now I thought eigenvectors and eigenvalues are something that's related to linear transformations. The said matrix is not one of any linear transformation. What do I do?
I need help with an integral eigenvalue equation...I am lost on how to handle this:
\int_{-\infty}^{\infty} dy K(x,y) \psi_n(y) = \lambda_n \psi_n(x)
The kernel, K(x,y) is a 2D, correlated Gaussian. I have read that for this case an analytic solution exist for the eigenvalues, \lambda_n...
I need to find the eigenvalues and eigenvectors of a matrix of the form
\left ( \begin{array}{cc}
X_1 & X_2 \\
X_2 & X_1
\end{array} \right )
where the X_i's are themselves M \times M matrices of the form
X_i = x_i \left ( \begin{array}{cccc}
1 & 1 & \cdots & 1 \\
1 & 1 & \cdots &...
if you have a differential equation of the form
x' = Ax
where A is the coefficient matrix, and you get a triple eigenvalue with a defect of 1. (meaning you get v1 and v2 as the associated eigenvector). How do you get v3 and how do you set up the solutions?
I tried finding v3 such that...
Let A be an nxn mx with n distinct eigenvalues and let B be an nxn mx with AB=BA. if X is an eigenvector of A, show that BX is zero or is an eigenvector of A with the same eigenvalue. Conclude that X is also an eigenvector of B.
I could show BX is zero or is an eigenvector of A with the...
I've got a homework problem that I am needing to do; however, I am not sure really what the question is asking. Obviously since I don't know what is being asked, I don't know where to begin.
I was hoping for some insight.
Question:
Show that matrix
A = {cos (theta) sin (theta), -sin...
I have a question that deals with all three of the terms in the title. I'm not really even sure where to begin on this. I was hoping someone could help.
Question:
An n x n matrix A is said to be idempotent if A^2 = A. Show that if λ is an eigenvalue of an independent matrix, then λ must...
I have two excercises which have been causing me to tear my hair off for some time now.
(a) the power method to find largest eigenvalue of A is defined as x(k+1) = Ax(k)
(b) the inverse power method is to solve Ax(k+1) = x(k) to find smallest eigenvalue of A
(c) the smallest/largest...
Hi,
How can you infer from these equations,
a = b_{max}(b_{max}+\hbar) \quad \text{and} \quad a = b_{min}(b_{min}-\hbar),
that b_{max} = -b_{min}? It is used in the derivation of the angular momentum eigenvalues...
I have the tridiagonal matrix (which comes from the backward Euler scheme)
A =
[ 1+2M - M 0 ... ]
[ -M 1+2M 0 ... ]
[ ... ]
[ -M 1+2M ]
I am given that the...
Hi, I'm wondering if there is some kind of shortcut for finding the eigenvalues and eigenvectors of the following matrix.
C = \left[ {\begin{array}{*{20}c}
{0.8} & {0.3} \\
{0.3} & {0.7} \\
\end{array}} \right]
Solving the equation \det \left( {C - \lambda I} \right) = 0, I...
confused on finding Eigenvalues and Eigenvectors!
hello everyone, i can't understand this example, how did they find the Eigen value of 3?! Aslo an Eigen vector of 1 1? http://img438.imageshack.us/img438/1466/lastscan1oc.jpg
thanks.
Hi I'm stuck on the following question and I have little idea as to how to proceed.
Note: I only know how to calculate eigenvalues of a matrix, I don't many applications of them(apart from finding powers of matrices). Also, I will denote the inner product by <a,b> rather than with circular...
I'm experiencing difficulties trying to find the eigenvalues of the follow matrix. The hint is to use an elementary row operation to simplify C - \lambda I but I can't think of a suitable one to use or figure out whether a single row operation will actually make the calculations simpler.
C...
howdy all,
i need some answers if possible
suppose i have a particle mass m, confinded in a 3d box sides L,2L,2L
what would be the energy eigenvalues of this particle
i presumed it to be:
hcross*w*A
where hcross is h/2*pi
w is omega
and A is the...
This is probably a straight forward question, but can someone show me how to solve this problem:
\frac {d^2} {d \phi^2} f(\phi) = q f(\phi)
I need to solve for f, and the solution indicates the answer is:
f_{\substack{+\\-}} (\phi) = A e^{\substack{+\\-} \sqrt{q} \phi}
I know...
1). suppose that y1, y2, y3 are the eigenvalues of a 3 by 3 matrix A, and suppose that u1, u2,u3 are corresponding eigenvectors. Prove that if { u1, u2, u3 } is a linearly independent set and if p(t) is the characteristic polynomial for A, then p(A) is the zero matrix.
I thought...
i'm reading and doing some work in introduction to linear algebra fifth edition, and i came across some problems that i had no clue.
1. An (n x n) matrix A is a skew symmetric (A(transposed) = -A). Argue that an (n x n) skew-symmetrix matrix is singular when n is an odd integer.
2. Prove...
This thread, https://www.physicsforums.com/showthread.php?t=74810, was orignally posted here in the QM forum, but it was moved to the homework section, which is reasonable. But nobody there knows quantum mechanics. I guess the OP gave up on it, but I'm curious how to do the problem now. So if...
Hi. I have this problem which i am stuck at:
Consider a one-dimensional Hamilton operator of the form
H = \frac{P^2}{2M} - |v\rangle V \langle v|
where the potential strength V is a postive constant and |v \rangle\langle v| is a normalised projector, \langle v|v \rangle = 1 ...
If L^2 |f> = k^2 |f>, where L is a linear operator, |f> is a function, and k is a scalar, does that mean that L|f> = +/- k |f>? How would you prove this?
In a recent thread
https://www.physicsforums.com/showthread.php?t=67366
matt and cronxeh seemed to imply that we should all know that the product of the eigenvalues of a matrix equals its determinant. I don't remember hearing that very useful fact when I took linear algebra (except in the...
Hi,
I need help on these questions for an assignment. I've been working on them for a couple of days and not getting anywhere. Any help would be appreciated...
1) A certain 4X4 real matrix is known to have these properties:
1. Two fo the eigenvalues of A are 3 and 2
2. the number 3 is an...
I having trouble finding the eigenvalues and eigenfunctions for the operator
\hat{Q} = \frac{d^2}{d\phi^2},
where \phi is the azimuthal angle. The eigenfunctions are periodical,
f(\phi) = f(\phi + 2\pi),
which I think should put some restrictions on the eigenvalues.
I think...
Hello:
-was solving for the eigenvalues of a matrix. Obtained:
\lambda = 1 \pm 2i
-substituted back into matrix to try and solve for the eigenvectors:
\left(\begin{array}{cc}2-2i & -2\\4 & -2-2i\end{array}\right) \left(\begin{array}{cc}x_1 \\ x_2 \end{array}\right) = \mathbf{0}...
let A be a diagonalizable matrix with eignvalues = x1, x2, ..., xn
the characteristic polynomial of A is
p (x) = a1 (x)^n + a2 (x)^n-1 + ...+an+1
show that inverse A = q (A) for some polynomial q of degree less than n
I'm having trouble getting started on this problem... I just really don't understand what to do.
Solve
X'+2X'+(\lambda-\alpha)X=0, 0<x<1
X(0)=0
X'(1)=0
a. Is \lambda=1+\alpha an eigenvalue? What is the corresponding eigenfunction?
b. Find the equation that the other eigenvalues...
I'm trying to find the basis for a particular matrix and I get a 3 eigenvalues with two of them being identical to each other. What do I do to find the basis for the repeated eigenvalue? Will it have the same basis as the original number?
Thanks!
I'm reading an introductionary text on quantum physics and am stumbling a bit with the terms used.
The text discusses a finite potential box (one dimension, time independent). It calculates the conditions for the solutions of the wave functions, which I can follow perfectly.
At that point...
let's say..
there is a particle, with mass m, in a 2-dimensions x-y plane. in a region
0 < x < 3L ; 0 < y < 2L
how to calculate the energy eigenvalues and eigenfunctions of the particle?
thx :smile:
and.. 2nd question..
there is a particle of kinetic energy E is incident from...
Who knows the formula to calculate the eigenvalues of total angular momentum between two different states? In particular, what is the matrix element of
<S, L, J, M_J | J^2 | S', L', J', M'_J> ?
Thank's...
spanning sets, eigenvalues, eigenvectors etc...
can anyone please explain to me what a spanning set is? I've been having some difficulty with this for a long time and my final exam is almost here.
also, what are eigenvalues and eigenvectors? i know how to calculate them but i don't understand...
Find the eigenvalues and eigenvectors of the general real symmetric 2 x 2 matrix A= a b
b c
The two eigenvalues that I got are a-b and c-b. I got these values from this:
(a-eigenvalue)(c-eigenvalue)-b^2=0
(a-eigenvalue)(c-eigenvalue) = b^2
(a-eigenvalue)= b = a-b...
I'm currently taking linear algebra and it has to be the worst math class EVER. It is extremely easy, but I find the lack of application discouraging. I really want to understand how the concepts arose and not simple memorize an algorithm to solve mindless operations, which are tedious. My...
I'm trying to show the relation between L^2 and Lz where L is total angular momentum and Lz is the z component.
Given f is an eigenfunction of both L^2 and Lz
L^2f = [lamb] f Lz f = [mu] f and L^2 = Lx^2 + Ly^2 + Lz^2 then
<L^2> = < Lx^2 + Ly^2 + Lz^2> = <Lx^2> + <Ly^2> + <Lz^2>...
the signature of a metric is often defined to be the number of positive eigenvalues minus negative eigenvalues of the metric.
this definition has always seemed a little suspicious to me. eigenvalues are defined for endomorphisms of a linear space, whereas the metric is a bilinear functional...