Eigenvectors Definition and 452 Threads

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...

Hi, I am trying to prove that the eigevalues, elements, eigenfunctions or/and eigenvectors of a matrix A form a Hilbert space. Can one apply the inner product formula : \begin{equation} \int x(t)\overline y(t) dt \end{equation} on the x and y coordinates of the eigenvectors [x_1,y_1] and...

Hello, In the case of 2D vector spaces, every vector member of the vector space can be expressed as a linear combination of two independent vectors which together form a basis. There are infinitely many possible and valid bases, each containing two independent vectors (not necessarily...

Homework Statement The Hamiltonian of a certain two-level system is: $$\hat H = \epsilon (|1 \rangle \langle 1 | - |2 \rangle \langle 2 | + |1 \rangle \langle 2 | + |2 \rangle \langle 1 |)$$ Where ##|1 \rangle, |2 \rangle## is an orthonormal basis and ##\epsilon## is a number with units of...

Homework Statement Coupled Harmonic Oscillators. In this series of exercises you are asked to generalize the material on harmonic oscillators in Section 6.2 to the case where the oscillators are coupled. Suppose there are two masses m1 and m2 attached to springs and walls as shown in Figure...

Homework Statement Prove that if the eigenvalues of a matrix A are \lambda_1 ... \lambda_n with corresponding eigenvectors x_1...x_n then \lambda^m_1...\lambda^m_n are eigenvalues of A^m with corresponding eigenvectors x_1...x_n Homework Equations Ax= \lambda x The Attempt at a...

Hi, I'm trying to calculate the eigenvectors of a 4x4 matrix, but I don't want the actual eigenvalues included in the solution, I simply want them listed as a variable. For example, I have the matrix: H_F = \left[ \begin{array}{cccc} \hbar\Omega&\hbar v_fk_- &0&0\\ \hbar...

I'm trying to recreate some results from a paper: https://arxiv.org/pdf/1406.1711.pdf Basically they take the Hamiltonian of graphene near the Dirac point (upon irradiation by a time periodic external field) and use Floquet formalism to rewrite it in an extended Hilbert space incorporating...

Homework Statement ##H=\frac{J}{4}\sum_{i=1}^2 \sigma_i^x \sigma_{i+1}^x## Homework Equations ##\sigma^x ## is Pauli matrix and ##J## is number.[/B]The Attempt at a Solution For ##i=1## to ##3## what is dimension of eigen vector? I think it is ##8##. Because it is like that I have tri sites...

Homework Statement Let ##T## be a linear operator on a vector space ##V##, and let ##\lambda_1,\lambda_2, \dots, \lambda_n## be distinct eigenvalues of ##T##. If ##v_1, v_2, \dots , v_n## are eigenvectors of ##T## such that ##\lambda_i## corresponds to ##v_i \ (1 \le i \le k)##, then ##\{ v_1...

Let's say we have a set of eigenvectors of a certain n-square matrix. I understand why the vectors are linearly independent if each vector belongs to a distinct eigenvalue. However the set is comprised of subsets of vectors, where the vectors of each subset belong to the same eigenvalue. For...

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...

Is it possible to find matrices that commute but eigenvectors of one matrix are not the eigenvectors of the other one. Could you give me example for it?

Homework Statement Consider a particle with angular momentum l=1. Write down the matrix representation for the operators L_x,\,L_y,\,L_z,for this particle. Let the Hamiltonian of this particle be H = aL\cdot L-gL_z,\,g>0.Find its energy values and eigenstates. At time t=0,we make a measurement...

Homework Statement T/F: Each eigenvector of an invertible matrix A is also an eignevector of A-1 Homework EquationsThe Attempt at a Solution I know that if A is invertible and ##A\vec{v} = \lambda \vec{v}##, then ##A^{-1} \vec{v} = \frac{1}{\lambda} \vec{v}##, which seems to imply that A and...

1. 1) Given 2x2 matrix A with A^t = A. How many linearly independent eigenvectors is A? 2) Is a square matrix with zero eigenvalue invertible? 2; When it comes to whether it is invertible; the det(A-λ* I) v = 0 where det (A-λ * I) v = 0 where λ = 0 We get Av = 0, where the eigenvector is...

Consider the eigenvectors ##(0, 1)## and ##(1, 0)## for the quantum system described the magnetic field ##\vec{B} = (0,0,B)##. Say I now rotate the magnetic field as ##\vec{B} = (B\sin\theta\cos\phi,B\sin\theta\sin\phi,B\cos\theta)##. Then the eigenvectors are supposed to change as...

Hello everybody, From a complete set of orthogonal basis vector ##|i\rangle## ##\in## Hilbert space (##i## = ##1## to ##n##), I construct and obtain a nondiagonal Hamiltonian matrix $$ \left( \begin{array}{cccccc} \langle1|H|1\rangle & \langle1|H|2\rangle & . &. &.& \langle1|H|n\rangle \\...

I am looking at some notes on Linear algebra written for maths students mainly to improve my Quantum Mechanics. I came across the following example - $$ \begin{pmatrix} 2 & -3 & 1 \\ 1 & -2 & 1 \\ 1 & -3 & 2 \end{pmatrix} $$ The example then gives the eigenvalues as 0 and 1(doubly degenerate)...

If i have an arbitrary ket then i know it can always be expressed as a linear combination of the basis kets.I now have an operator A which has 2 eigenvalues +1 and -1. The corresponding eigenvectors are | v >+ = k | b > + m | a > and | v >- = n | c > where | a > , | b > and | c > are...

When considering the 2 eigenvectors of the Sz operator the | + > eigenvector points in the positive z direction and the | - > points in the negative z direction ; so is it correct to write | + > = - | - > ? And similarly for the eigenvectors of the Sx and Sy operators ? Thanks

I am new to linear algebra but I have been trying to figure out this question. Everybody seems to take for granted that for matrix A which has eigenvectors x, A2 also has the same eigenvectors? I know that people are just operating on the equation Ax=λx, saying that A2x=A(Ax)=A(λx) and...

Homework Statement Homework Equations The lattice laplacian is defined as \Delta^2 = \frac{T}{\tau} , where T is the transition matrix \left[ \begin{array}{cccc} -2 & 1 & 0 & 0 \\ 1 & -2 & 1 & 0 \\ 0 & 1 & -2 & 1 \\ 0 & 0 & 1 & -2 \end{array} \right] and \tau is a time constant, which is...

Let's say i have the 3x3 matrix a 0 0 b 0 0 1 2 1 it's eigenvalues are e1 =a, e2 = 0, e3 = 1. now if a = / = 0, 1 i have 3 distinct eigenvalues and the matrix is surely can be Diagonalizable. so if i try to solve for the eigenvector for the eigenvalue e1 =a: 0 0 0 b -a 0 1 2 1-a...

Homework Statement we have this matrix 6 - 1 0 -1 -1 -1 0 -1 1 We need to find it's eigenvalues and eigenvectors Homework Equations The Attempt at a Solution[/B] I wrote the characteristic equation - det(A- λxunit matrix) to find the roots and got (-λ^3)+8(λ^2)+λ-6 instead of...

I'm looking for the general form of a symmetric 3×3 matrix (or tensor) ##\textbf{A}## with only two different eigenvalues, i.e. of a matrix with the diagonalized form ##\textbf{D}=\begin{pmatrix}a& 0 & 0\\0 & b & 0\\0 & 0 & b\end{pmatrix} = \text{diag}(a,b,b)##. In general, such a matrix can be...

Homework Statement Okay this is the problem it seems so easy but i just cannot for the life of me get it to click into my mind, I have 4 unknowns and 5 equations and i have to put it into a matrix and try solve it matricies or eigenvalues/eigenvectors. The 5 equations are: a= b/2 b=a/3 + d...

Hey so I'm new to my TI nspire cx, still getting the hang of it. I've been trying to figure out how to get my eigenvector values to be fractions instead of decimals when I calculate them on here. Also, when I find the polynomial roots I get back decimals instead of fractions. I would like to...

This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf Given the operator \hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right) I find correctly that the eigenvalues are \lambda = b, \pm b. To find the eigenvectors for b, I do the following...

Homework Statement So I have been having trouble with finding the proper eigen vector for a complex eigen value for the matrix A=(-3 -5) . .....(3 1) had a little trouble with formating this matrix sorry The eigen values are -1+i√11 and -1-i√11 The Attempt at a Solution using AY-λY=0...

Homework Statement Find the eigenvalues and associated eigenvector of the following matrix: Homework EquationsThe Attempt at a Solution We have a theorem in our lectures notes that states that if a matrix is invertible the only eigenvector in its kernel will be the zero vector. In order...

Suppose ##v_i## is an eigenvector of ##A## with eigenvalue ##\lambda_i## and multiplicity ##1##. ##AA^{-1}v_i=A^{-1}Av_i=A^{-1}\lambda_iv_i=\lambda_iA^{-1}v_i## Thus ##A^{-1}v_i## is also an eigenvector of ##A## with the same eigenvalue ##\lambda_i##. Since the multiplicity of ##\lambda_i##...

If I have a system where the following is found to describe the motion of three particles: The normal modes of the system are given by the following eigenvectors: $$(1,0,-1), (1,1,1), (1,-2,1)$$ How can I find the corresponding eigenfrequencies? It should be simple... What am I missing?

Homework Statement Prove that the states $$|z, \alpha \rangle = \hat S(z)\hat D(\alpha) | 0 \rangle $$ $$|\alpha, z \rangle = \hat D(\alpha) \hat S(z)| 0 \rangle $$ are eigenvectors of the squeezed amplitude operator $$ \hat b = \hat S(z) \hat a \hat S ^\dagger (z) = \mu \hat a + \nu \hat a...

Homework Statement So just curious about a specific problem that I am worries about running into on my test tomorrow. When trying to find eigen vectors with the eigen values what is there is a discrepancy between the two systems obtained after doing the matrix arithmetic? such as after using...

Effie has correctly found that the eigenvalues of $\displaystyle \begin{align*} A = \left[ \begin{matrix} \phantom{-}3 & \phantom{-}2 \\ -3 & -4 \end{matrix} \right] \end{align*}$ are $\displaystyle \begin{align*} \lambda_1 = -3 \end{align*}$ and $\displaystyle \begin{align*} \lambda_2 = 2...

Homework Statement I have a spin operator and have to find the eigenstates from it and then calculate the eigenvalues. I think I managed to get the eigenvalues but am not sure how to get the eigenstates.Homework Equations The Attempt at a Solution I think I managed to get the eigenvalues out...

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...

Mark44 submitted a new PF Insights post What Are Eigenvectors and Eigenvalues? Continue reading the Original PF Insights Post.

Homework Statement X= 1st row: (0, 1, 0, 0), 2nd row: (1, 0, 0, 0), 3rd row: (0, 0, 0, 1-i), 4th row: (0, 0, 1+i, 0) Find the eigenvalues and eigenvectors of the matrix X. Homework Equations |X-λI|=0 (characteristic equation) (λ is the eigenvalues, I is the identity matrix) (X-λI)V=0 (V is the...

Is there a general between the eigenvectors of a matrix and the row (or column) vectors making up the matrix?

Homework Statement Given the linear transformation l : R 2 → R 2 defined below, find characteristic equation, real eigenvalues and corresponding eigenvectors. a) l(x, y) = (x + 5y, 2x + 4y) Homework Equations characteristic equation = det (A-λI) = 0 The Attempt at a Solution l(x, y) = (x +...

Hello all I have a theoretical question. I know how to find the eigenvalues and eigenvectors of a matrix A. What I am not sure about, is what it all means and why do we need it for. I did some reading, and saw something about stretching vector, if I not mistaken, if I have a vector v, and I...

Homework Statement A = \begin{bmatrix} 2 & 1 & 0\\ 0& -2 & 1\\ 0 & 0 & 1 \end{bmatrix} Homework EquationsThe Attempt at a Solution The spectrum of A is \sigma (A) = { \lambda _1, \lambda _2, \lambda _3 } = {2, -2, 1 } I was able to calculate vectors v_1 and v_3 correctly out of the...

I'm almost there in terms of understanding it, but I need to go beyond the text. Here is the example problem: imgur link: http://i.imgur.com/UMj55tF.jpg I can see that where we have 1 = \vec{x}^T A \vec{x} = \lambda \vec{x}^T \vec{x} that 1=\lambda \vec{x}^T \vec{x} = \lambda ||\vec{x}||^2...

In my text, it tells me to find the eigenvectors of a 2nd difference matrix and graph the eigenvectors to see how they fall onto sine curves. imgur link: http://i.imgur.com/oxbkTn6.jpg My question is simple but general. What does this even mean? How did they produce this graph from the...

Given a Positive Definite Matrix ## A \in {\mathbb{R}}^{2 \times 2} ## given by: $$ A = \begin{bmatrix} {A}_{11} & {A}_{12} \\ {A}_{12} & {A}_{22} \end{bmatrix} $$ And a Matrix ## B ## Given by: $$ B = \begin{bmatrix} \frac{1}{\sqrt{{A}_{11}}} & 0 \\ 0 & \frac{1}{\sqrt{{A}_{22}}}...

The problem is here, I'm trying to solve (b): imgur link: http://i.imgur.com/ifVm57o.jpg and the text solution is here: imgur link: http://i.imgur.com/qxPuMpu.pngI understand why there is a term in there with cte^t, it's because the A matrix has double roots for the eigenvalues. What I...

I thought I understood how to solve these sorts of equations, but apparently not.. 1. Homework Statement In Linear Algebra I'm solving diff eqs with eigenvectors to get all the combinations that will solve for a diff eq. The text then asked me to check my answer by going back and solving...