Eigenvectors Definition and 462 Threads

Hi everybody.. How can i use fortran99 to find the eigenvalues & eigenvectors of sparse matrices? Thanx :)

I considered the covariance of 2 spin 1/2 as a non linear operator : A\otimes B-A|\Psi\rangle\langle\Psi|B. The eigenvectors are but non orthogonal and I wondered what happens in that case with the probabilities : from Born"s rule it comes that the transition probability from one vector to the...

Hey everyone. I have a matrix A = {{7,-5,0},{-5,7,0},{0,0,-6}} I have found the Eigenvalues, 2,12,-6 but I'm only getting one Eigenvector, (0,0,1).. I know there is 2 others (-1,1,0) and (1,1,0) but I am unable to get them by hand. Once I get the matrix in the form (A-λI)*v = o, I just get...

Dear all, Recenty,I am reading the source code of the first-principle software.I meet some words that I haven't found in those DFT books.For example,it mentions the first-variational and second-variational eigenvectors. Similarly,the first-variational and second-variational eigenvalues are...

I know eigenvectors corresponding to different eigenvalues are linearly independent but what about a set ${e_{1},...,e_{n}}$ of eigenvectors corresponding to different eigenvalues?

hello How to you rigorously express the orthonormality of a complete set of eigenvectors (|q\rangle)_q of the position operator given that these are necessarily generalized eigenvectors (elements of the distribution space of a rigged hilbert space)? The usual unformal condition \langle...

Homework Statement Part (a): Find the eigenvalues and eigenvectors of matrix A: \left( \begin{array}{cc} 2 & 0 & -1\\ 0 & 2 & -1\\ -1 & -1 & 3 \\ \end{array} \right) Part(b): Find the eigenvalues and eigenvectors of matrix ##B = e^{3A} + 5I##. Homework Equations The Attempt at a Solution...

Hello, Homework Statement I want to show that a real symmetric matrix will have real eigenvalues and orthogonal eigenvectors. $$ \begin{pmatrix} A & H\\ H & B \end{pmatrix} $$ The Attempt at a Solution For the matrix shown above it's clear that the charateristic equation will be...

Homework Statement Show there are no eigenvectors of a^{\dagger} assuming the ground state |0> is the lowest energy state of the system. Homework Equations Coherent states of the SHO satisfy: a|z> = z|z> The Attempt at a Solution Based on the hint that was given (assume there...

Homework Statement Let T: R^6 -> R^6 be the linear operator defined by the following matrix(with respect to the standard basis of R^6): (0 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ) a) Find the T-cyclic subspace generated by each standard basis vector...

Homework Statement Given A = [ (3,-7),(1,-2) ] and λa = \frac{1}{2} + i \frac{\sqrt{3}}{2} find a single eigenvector which spans the eigenspace. Homework Equations The Attempt at a Solution So I row reduced the matrix to get [(2, -5 + i\sqrt{3}),(0,0 ] and from here we can...

Let A be an 3x3 matrix such that A\mathbf{v_1}=\mathbf{v_1}+\mathbf{v_2}, A\mathbf{v_2}=\mathbf{v_2}+\mathbf{v_3}, A\mathbf{v_3}=\mathbf{v_3} where \mathbf{v_3} \neq \mathbb{0}. Let B=S^{-1}AS where S is another 3x3 matrix. (i) Find the general solution of \dot{\mathbf{x}}=B\mathbf{x}. (ii)...

Homework Statement In a given basis, the eigenvectors A and B are represented by the following matrices: A = [ 1 0 0 ] B = [ 2 0 0 ] [ 0 -1 0] [ 0 0 -2i ] [ 0 0 -1] [ 0 2i 0 ] What are A and B's eigenvalues? Determine [A, B]. Obtain a set...

Hello guys, is there any way someone can explain to me in resume what eigen values and eigenvectors are because I don't really recall this theme from linear algebra, and I'm not getting intuition on where does Fourier transform comes from. my teacher wrote: A\overline{v} = λ\overline{v}...

Hello MHB, I got one question. If I want to find basis ker and it got double root in eigenvalue but in that eigenvalue i find one eigenvector(/basis) what kind of decission can I make? Is it that if a eigenvalue got double root Then it Will ALWAYS have Two eigenvector(/basis)? Regards, |\pi\rangle

Hi I was learning about eigenvectors, inner products, Dirac notation etc. But I don't get why wave functions are represented as eigenvectors?

The det. of the following matrix: $$ \begin{matrix} 2k-ω^{2}m_{1} & -k\\ -k & k-ω^{2}m_{2}\\ \end{matrix} $$ must be equal to 0 for there to be a non-trivial solution to the equation: $$(k - ω^{2}m)x =0$$ Where m is the mass matrix: $$ \begin{matrix} m_{1} & 0\\ 0& m_{2}\\...

Hi everyone, :) Here's another question that I solved. Let me know if you see any mistakes or if you have any other comments. Thanks very much. :) Problem: Prove that the eigenvector \(v\) of \(f:V\rightarrow V\) over a field \(F\), with eigenvalue \(\lambda\), is an eigenvector of \(P(f)\)...

Hi you all. I have to diagonalize a hermitian operator (hamiltonian), that has both discrete and continuous spectrum. If ψ is an eigenvector with eigenvalue in the continuous spectrum, and χ is an eigenvector with eigenvalue in the discrete spectrum, is correct to say that ψ and χ are always...

Hi guys! I need help on a problem from one of my GR texts. Suppose that ##\xi^a## is a killing vector field and consider its twist ##\omega_a = \epsilon_{abcd}\xi^b \nabla^c \xi^d##. I must show that ##\omega_a = \nabla_a \omega## for some scalar field ##\omega##, which is equivalent to showing...

Homework Statement How can I produce a Lyapunov function using the eigenvalues and vectors x'=-x+y y'=-x Homework Equations The Attempt at a Solution So I got the matrix using jacobian and I got the matrix -1 1 -1 0 then i found the eigenvalues to be λ_1= (-1+sqrt3...

Dear All, In general eigenvalue problem solutions we obtain the eigenvalues along with eigenvectors. Eigenvalues are unique for each individual problem but eigenvectors are not, since the case is like that how we can rely that solution based on the eigenvector is correct. Because if solution is...

Homework Statement Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could...

Good evening :-) I have an exam on Wednesday and am working through some past papers. My uni doesn't give the model answers out, and I have come a bit stuck with one question. I have done part one, but not sure where to go from here, would be great if someone could point me in the right...

can someone PLEASE explain eigenvalues and eigenvectors and how to calculate them or a link to a site that teaches it simply?

Homework Statement Hi so I have the eigenvalue equation S\vec x = λ\vec x where S = \frac{\hbar}{2}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} I have correctly calculated the eigenvalues to be λ=±\frac{\hbar}{2} and the corresponding normalised eigenvectors to be: \hat{e}_{1}=...

Hey guys if i have a vector x=[x1,x2, ... xn] what are the eigenvectors and eigenvalues of X^T*X ? I know that i get a n by n symmetric matrix with it's diagonal entries in the form of Ʃ xii^2 for i=1,2,3,. . . ,n Thank you in advance once again!

Homework Statement Find the eigenvalues and eigenvectors of P = {(0.8 0.6), (0.2 0.4)}. Express {(1), (0)} and {(0), (1)} as sums of eigenvectors. Homework Equations Row ops and det(P - λI) = 0. The Attempt at a Solution I've found the eigenvectors and eigenvalues of P to be 1...

Hi, can someone help me about rotating phonon eigenvectors? Say I have a primitive cell with 3 atoms, so for each point in reciprocal space, there are 9 eigen frequency and eigenvectors. Each eigenvector is a 9-dimensional complex vector. I calculated eigenvectors from a transverse branch...

So I am given B=\begin{array}{cc} 3 & 5 \\ 5 & 3 \end{array}. I find the eigenvalues and eigenvectors: 8, -2, and (1, 1), (1, -1), respectively. I am then told to form the matrix of normalised eigenvectors, S, and I do, then to find S^{-1}BS, which, with S = \frac{1}{\sqrt{2}}\begin{array}{cc} 1...

Here is the question: Here is a link to the question: How to I find corresponding eigenvectors? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement I'm given this matrice 2 1 0 1 2 0 0 0 3 and I need to find it's eigenvectors Homework Equations The Attempt at a Solution So I get the eigenvalues to be 1,3,3 with 3 being the one with...

Hi, Does anyone know how to prove that two commutative Hermitian matrices can always have the same set of eigenvectors? i.e. AB - BA=0 A and B are both Hermitian matrices, how to prove A and B have the same set of eigenvectors? Thanks!

Homework Statement This is my first post, so forgive me if anything's out of order. Assumean operator A satisfies the following equation: 1+2A-A^2-2A^3=0 Find the eigenvalues and eigenvectors for A Homework Equations The Attempt at a Solution So the eigenvalues are +1,-1, and...

Hi, I have a quick question, when should one use RREF in finding eigvenvectors? I've read through some books and sometimes they use them and sometimes they do not. I'm sorry for such a potentially stupid question. Thanks, Zubin

If you solve the Schrödinger equation time independent and find a number of stationary position states they are eigenstates. So say uou find the eigen state ψ then c*ψ is also an eigenstate, Does it matter which of these I pick as the eigenstate or is it only the eigenvalue that matters?

Hi guys, in my college we have this "Independent Study" component for the Intro Linear Algebra class. Basically, I have to solve 4 questions that go beyond what was covered in class. Sadly, being the procrastinator I am, I haven't looked at it until now and the assignment is due Friday. The...

Hallo, I am trying to solve the following problem. I need to get eigenvectors of a matrix. I know that there are many subroutines for that in linear algebra packages, for instance in Lapack there is DSPEV, but they all give normalized eigenvectors, while I need the "original" unnormalized ones...

Homework Statement operators: K=LM and [L,M]=1 α is an eigenvector of K with eigenvalue λ. Show that x=Lα and y=Mα are also eigenvectors of K and also find their eigenvalues. Homework Equations K=LM [L,M]=1 Kα=λα The Attempt at a Solution I tried, but its not even worth...

Homework Statement Use Lagrange multipliers to find the eigenvalues and eigenvectors of the matrix A=\begin{bmatrix}2 & 4\\4 & 8\end{bmatrix} Homework Equations ... The Attempt at a Solution The book deals with this as an exercise. From what I understand, it says to consider...

Matrix A= 2 1 2 1 2 -2 2 -2 -1 It's known that it has eigenvalues d1=-3, d2=d3=3Because it has 3 eigenvalues, it should have 3 linearly independent eigenvectors, right? I tried to solve it on paper and got only 1 linearly independent vector from d1=-3 and 1 from d2=d3=3. The method I used...

In Quantum, I ran across the eigenvalue problem. They gave me a matrix, and i was asked to find eigenvalues and then eigenvectors. But the eigenvalues, were degenerate and thus i couldn't find the exact normalized eigenvector. What to do in this case? Shoukd i choose arbitrary values? My...

Hi We have a matrix A (picture), the eigenvalues are λ1 = 4 and λ2 = 1 and the eigenvectors are λ1 : t(1,0,1) λ2 : t1(1,0,2) + t2(0,1,0) I have to examine if there's a column vector v that satifies : A*v = 2 v I would say no there doesn't exist such a column vector v because 2 isn't an...

Homework Statement An observable is represented by the matrix 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 \frac{1}{\sqrt{2}} 0 Find the normalized eigenvectors and corresponding eigenvalues. The Attempt at a Solution I...

I got 2 questions about eigenvectors. Let's say you have an eigenvector [1 0 2]^t. 1. Does the order matter? Like can I change the order to [0 1 2]^t or [1 2 0]^t? 2. It can be any scalar multiple of the vector right? Like I could have [2 0 4] or [4 0 8]

Hi all Homework Statement Given is a Hermitian Operator H H= \begin{pmatrix} a & b \\ b & -a \end{pmatrix} where as a=rcos \phi , b=rsin \phi I shall find the Eigen values as well as the Eigenvectors. Furthermore I shall show that the normalized quantum states are: \mid +...

Homework Statement A matrix A has eigenvectors [2,1] [1,-1] and eigenvalues 2 , -3 respectively. Determine Ab for the vector b = [1,1]. Homework Equations The Attempt at a Solution First I put be as a combination of the two eigenvectors ie 2/3[2,1] -1/3[1,-1] = b...

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...

I am wondering if there is a systematic way to fix the phase of complex eigenvectors. For example e^{i \theta}(1,\omega,\omega^2) where e^{i \theta} is an arbitrary phase and \omega and \omega^2 are the cube roots of unity, is an eigenvector of the cyclic matrix \left(\begin{matrix}0&...

-2x + 3y + z = 0 3x + 4y -5z = 0 x -2y + z = -4 Find the characteristic equation, eigenvalues / eigenvectors of the system. I'm given to understand the eigenvalue problem is Ax = (lamba)x, but lamba doesn't exist in the system above. How can I solve for the eigenvalues when there are none?