Eigenvectors Definition and 462 Threads

The eigenvectors of a hermitian matrix corresponding to unique eigenvalues are orthogonal. This is not too difficult of a statement to prove using mathematical induction. However, this case is seriously bothering me. Why is the dot product of the vectors not rightly zero? Is there something more...

Why are the eigenvectors of this hermitian matrix not checking out as orthogonal? The eigenvalues are certainly distinct. ChatGPT also is miscalculating repeatedly. I have checked my work many times and cannot find the error. Kindly assist.

Hi all, (first post here :D) I am working on periodic dielectric structures in the long-wavelength limit (wavelength much larger than the periodicity). In the long wavelength limit the periodic strucutre can be homogonized and described via an effective permittivity (or refractive index)...

Hey guys, I was wondering if anyone had any thoughts on the following symmetric matrix: $$\begin{pmatrix} 0.6 & 0.2 & -0.2 & -0.6 & -1\\ 0.2 & -0.2 & -0.2 & 0.2 & 1\\ -0.2 & -0.2 & 0.2 & 0.2 & -1\\ -0.6 & 0.2 & 0.2 & -0.6 & 1\\ -1 & 1 & -1 & 1 & -1 \end{pmatrix} $$ Notably, when one solves for...

Before going through calculations/reasoning, let me summarize what my questions will be - In order to obtain the desired matrix, I impose five constraints on ##a,b,c,d,## and ##\lambda##. - These five constraints are four equations and an inequality. I am not sure how to work with the...

I tried to find the answer to this but so far no luck. I have been thinking of the following: I generate two random vectors of the same length and assign one of them as the right eigenvector and the other as the left eigenvector. Can I be sure a matrix exists that has those eigenvectors?

Hi; struggling a little with eigenvectors; I can get to the equation at the foot of the example but I can't understand the "formula" leading to the setting of x = 3 at the foot of the example? thanks martyn

For exercise 3 (2), , The solution for finding the eigenvector is, However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here? Many Thanks!

For this, The solution is, However, does someone please know what allows them to express the eigenvector for each of the sub-matrix in terms of t? Many thanks!

Assume a table A(3x3) with the following: A [ 1 2 1 ]^T = 6 [ 1 2 1 ]^T A [ 1 -1 1 ]^T = 3 [ 1 -1 1 ]^T A [ 2 -1 0]^T = 3 [ 1 -1 1]^T Find the Eigenvalues and eigenvectors: I have in mind to start with the Av=λv or det(A-λI)v=0.... Also, the first 2 equations seems to have the form Av=λv...

Hi, In my linear algebra homework, there is a bonus assignment where we are supposed to use Mathematica to calculate matrices and their determinants etc. here is the assignment. Unfortunately, I am a complete newbie when it comes to Mathematica, this is the first time I have worked with...

My understanding is: $$\phi (\mathbf{k})=\int{d^3}\mathbf{x}\phi (\mathbf{x})e^{-i\mathbf{k}\cdot \mathbf{x}}$$ But what is ##\phi (\mathbf{x})## in Qft? In quantum mechanics, $$|\phi \rangle =\int{d^3}\mathbf{x}\phi (\mathbf{x})\left| \mathbf{x} \right> =\int{d^3}\mathbf{k}\phi...

For this, Dose someone please know where they get P and D from? Also for ##M^k##, why did they only raise the the 2nd matrix to the power of k? Many thanks!

For this, Dose anybody please know why we cannot say ##\lambda = 1## and then ##1## would be the eigenvalue of the matrix? Many thanks!

In Cohen-Tannoudji page 423, they try to teach a method that allows to find the eigenvectors of a 2-state system in a less cumbersome way. I understand the steps, up to the part where they go from equation (20) to (21). I understand that (20) it automatically leads to (21). Can someone please...

Hi, I have to find the eigenvalue (first order) and eigenvector (0 order) for the first and second excited state (degenerate) for a perturbated hamiltonian. However, I don't see how to find the eigenvectors. To find the eigenvalues for the first excited state I build this matrix ##...

Hi, If ##|a\rangle## is an eigenvector of the operator ##A##, I know that for any scalar ##c \neq 0## , ##c|a\rangle## is also an eigenvector of ##A## Now, is the ket ##F(B)|a\rangle## an eigenvector of ##A##? Where ##B## is an operator and ##F(B)## a function of ##B##. Is there way to show...

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...

The statement " If ##T: V \to V## has the property that ##T^2## has a non-negative eigenvalue ##\lambda^2##", means that there exists an ##x## in ##V## such that ## T^2 (x) = \lambda^2 x##. If ##T(x) = \mu x##, we've have $$ T [T(x)]= T ( \mu x)$$ $$ T^2 (x) = \mu^2 x$$ $$ \lambda ^2 = \mu ^2...

So I have been studying for my upcoming math exam and a lot of the problems require to find eigenvalues/eigenvectors.Now the question I have is the following; Take a look at this matrix $$ \left[ \begin{matrix} 6 & -3 \\\ 3 & -4 \end{matrix} \right] $$ Now the eigenvalues are...

Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...

Hello everyone. I am trying to construct a functioning version of randomfields (specifically 2D_karhunen_loeve_identification_example.py) in Matlab. For that, I have to calculate the Karhunen-Loève expansion of 2D data, since this is what it says in the documentation. I also have some sample...

The J3 matrix of two dimensional SU2 consists of two row vectors (1 0) and (0 -1). When I calculate the eigenvalues of an eigenvector v the usual way with J3v=kv then I find eigenvalues +-1 and eigenvectors (1 0) and (0 1). But how is it possible to say that there are other eigenvectors and...

good evening everyone! Decided to solve the problems from last year's exams. I came across this example. Honestly, I didn't understand it. Who can help a young student? :) Find characteristic equation of the matrix A in the form of the polynomial of degree of 3 (you do not need to find...

In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...

For ##M = \begin{pmatrix} 2 & 2\\ 2 & -1 \end{pmatrix}## I found the characteristic equation: ##( λ - 3 )( λ + 2) \therefore λ = 3,-2##Going back we multiply $$\begin{pmatrix} 2 - \lambda & 2\\ 2 & -1 - \lambda \end{pmatrix}\begin{pmatrix} x\\ y \end{pmatrix}$$ Which gives \begin{matrix} 2x -...

Zach Star gives an explanation of Eigenvalues and Eigenvectors, with some applications

Hi, I just have a quick question when I was working through a linear algebra homework problem. We are given a matrix A = \begin{pmatrix} 2 & -2 \\ 1 & -1 \end{pmatrix} and are asked to compute e^{A} . In earlier parts of the question, we prove the identities A = V \Lambda V^{-1} and e^{A}...

upon finding the eigenvalues and setting up the equations for eigenvectors, I set up the following equations. So I took b as a free variable to solve the equation int he following way. But I also realized that it would be possible to take a as a free variable, so I tried taking a as a free...

From solving the characteristic equations, I got that ##\lambda = 0.5 \pm 1.5i##. Since using either value yields the same answer, let ##\lambda = 0.5 - 1.5i##. Then from solving the system for the eigenvector, I get that the eigenvector is ##{i}\choose{1.5}##. Hence the complex solution is...

Anyone know what result this article is talking about? https://www.theatlantic.com/science/archive/2019/11/neutrino-oscillations-lead-striking-mathematical-discovery/602128/

A link to an interesting article I found is below: https://www.quantamagazine.org/neutrinos-lead-to-unexpected-discovery-in-basic-math-20191113/

An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm...

Say I have a matrix ##A## and it has three eigenvectors ##|\psi_1\rangle##, ##|\psi_2\rangle## and ##|\psi_3\rangle##. I want to orthogonalize these. Say my orthogonalized eigenvectors are ##|\phi_1\rangle##, ##|\phi_2\rangle## and ##|\phi_3\rangle##. $$ \begin{eqnarray} |\phi_1\rangle =...

Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!

My question is that what is the physical meaning of the EM stress-energy tensor's eigenvectors? Thanks for the answers - Robert

Hello everyone. I am currently using the pca function from MATLAB on a gaussian process. Matlab's pca offers three results. Coeff, Score and Latent. Latent are the eigenvalues of the covariance matrix, Coeff are the eigenvectors of said matrix and Score are the representation of the original...

We know that S2 commutes with Sz and so they share their eigenspace. Now since S2 also commutes with Sx, as per my understanding, the eigenvectors of S2 and Sz should also be the eigenvectors of Sx. But since the paulic matrices σx and σy are not diagonlized in the eigenbasis of S2, it is clear...

Here's the problem along with the solution. The correct answer listed in the book for the eigenvectors are the expressions to the right (inside the blue box). To find the eigenvectors, I tried using a trick, which I don't remember where I saw, but said that one can quickly find eigenvectors (at...

Homework Statement Find the eigenvalues and eigenvectors of the following matrix: $$ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 2 \\ 0 & -1 & 0 \end{bmatrix} $$ Homework Equations Characteristic polynomial: $$ \Delta (t) = t^3 - Tr(A) t^2 + (A_{11}+A_{22} +A_{33})t - det(A) .$$ The Attempt at...

Homework Statement Find the eigenvalues and eigenvectors fro the matrix: $$ A=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} $$. Homework Equations Characteristic polynomial: ## \nabla \left( t \right) = t^2 - tr\left( A \right)t + \left| A \right|## . The Attempt at a Solution I've found...

Homework Statement I have got the following matrix. I have found the eigen values but in some eq x, y & z terms are vanishing, so how to find the value of eigen vector? Also why we have to do normalization?? A__=__[1__1__0] ______[1__1__0] ______[0__0__1]Homework Equations A-λI=0 Ax = -λIx...

Homework Statement I am continuing from : https://www.physicsforums.com/threads/finding-eigen-values-list-of-possible-solutions-for-lambda.955164/ I have got a 3 * 3 matrix. I have to find itseigen values and eigen vectors. I have found the eigen values.For calculating eigen vectors they are...

Homework Statement Consider the following Matrix: Row1 = 2 2 Row2 = 5 -1 Find its Eigen Vectors Homework Equations Ax = λx & det(A − λI)= 0. The Attempt at a Solution First find the det(A − λI)= 0. which gives a quadratic eq. roots are λ1 = -3 and λ2 = 4 (Eigen values) Then using λ1, I...

1. n = sinθcosφ i + sinθsinφ j + cos k σ = σx i + σy j + σz k , where σi is a Pauli spin matrix Find the eigen vectors for the operator σ⋅n 2. Determinant of (σ⋅n - λI), where I is the identity matrix, needs to equal zero (σ⋅n - λI)v = 0, where v is an eigen vector, and 0 is the zero vector...

Given matrices A,B and Condition 1: AB does not equal BA Condition 2: A and B do not have common eigenvectors are these two conditions equivalent? If not, exactly how are they related? Since I'm thinking about quantum mechanics I'm wondering specifically about Hermitian matrices, but I'm...

Hi, I am trying to find the eigenvectors for the following 3x3 matrix and are having trouble with it. The matrix is (I have a ; since I can't have a space between each column. Sorry): [20 ; -10 ; 0] [-10 ; 30 ; 0] [0 ; 0 ; 40] I’ve already...

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...

Homework Statement Find the eigenvalues and eigenvectors of the matrix ##A=\matrix{{2, 0, -1}\\{0, 2, -1}\\{-1, -1, 3} }## What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix?Homework EquationsThe Attempt at a Solution So I've found the...

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...