Matrices Definition and 1000 Threads

Hey! :giggle: The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following: (a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...

Hello there I am having trouble with part b) of this exercise. I can apply the rotation matrix easily enough and get: $$ R(-\theta) \vec J= \begin{bmatrix} A\cos\theta + B\sin{\theta}e^{i\delta} \\ -A\sin\theta + B\cos{\theta}e^{i\delta} \end{bmatrix} $$ I decided to convert the exponential...

$\tiny{311.1.1.26}$ Construct 3 augmented matrices for linear systems whose solution set is $x_1=3, \quad x_2=-2, \quad x_3=-1$ ok the only thing I could think of is just rearrange the rows of an RREF matrix. albeit losing the triangle format hopefully no typos $\left[\begin{array}{rrr|rr} 1&...

Hello, I want to ask if anyone can explain to me how to multiply indexed matrices. This is an example I have made, but I do not know if it is true

Summary:: Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix and let ##\lambda \in \Bbb R## be an eigenvalue of ##A##. Prove that the geometric multiplicity ##g(\lambda)## of ##A## equals its algebraic multiplicity ##a(\lambda)##. Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix...

Hello all, I am currently working on the four fundamental spaces of a matrix. I have a question about the orthogonality of the row space to the null space column space to the left null space ------------------------------------------------ In the book of G. Strang there is this nice picture...

Hello, I am studying change of basis in linear algebra and I have trouble figuring what my result should look like. From what I understand, I need to express the "coordinates" of matrix ##A## with respect to the basis given in ##S##, and I can easily see that ##A = -A_1 + A_2 - A_3 + 3A_4##...

I have already in a previous task shown that A is not irreducible and not regular, which I think is correct. I don't know if I can use that fact here in some way. I guess one way of solving this problem could be to find all eigenvalues, eigenvectors and diagonalize but that is a lot of work and...

Here is the paper again: https://www.mdpi.com/2218-2004/6/2/22?type=check_update&version=2#related_content For a class project I need to calculate the energy levels of atoms using the Hartree Fock method as presented in this paper which essentially brute forces the calculation using finite...

Hey! 😊 We have the matrices \begin{equation*}s:=\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}, \ d:=\frac{1}{2}\begin{pmatrix}-1 & -\sqrt{3} \\ \sqrt{3} & -1\end{pmatrix}\end{equation*} and the points \begin{equation*}p:=\begin{pmatrix}2 \\ 0 \end{pmatrix}, \ q:=\begin{pmatrix}-1 \\ \sqrt{3}...

I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...

I read that density matrices are useful in Physics mainly to describe a) mixtures; we do not know the wave function of the system so ##\psi## is random and b) entangled systems. I'd like to focus on the later. Let us have a system ##S_1## entangled to the system ##S_2##. Thus we start from a...

I thought i understood the theorem below: i) If A is a matrix in ##M_n(k)## and the minimal polynomial of A is irreducible, then ##K = \{p(A): p (x) \in k [x]\}## is a finite field Then this example came up: The polynomial ##q(x) = x^2 + 1## is irreducible over the real numbers and the matrix...

See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first. Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...

In my book no explanation for this concept is given and i can't find anything about it when I am searching. One example that was given was: Let $$A=\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}$$ with ##k=\mathbb{Z}_2## I think k is the set of scalars for a vector that can be multiplied with...

I have this set of adjacency matrices and networks given to me and I need to match the matrix to the network and then describe the dynamics. At first glance it seems to be Ac, Bb and Ca but I'm not sure if that's too obvious and I'm missing something as I haven't looked at visual...

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows are the same in the 2 matrixes). Now I would like to make the cross synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from Maximum Likelihood Estimator...

Hi! Is it possible to construct gamma matrices satisfying the Clifford algebra ##\{\gamma^\mu, \gamma^\nu \} = \eta^{\mu \nu}## that are *real*, for ##\eta = diag(-1,1,1)##? I know that I can construct them in principle from sigma matrices, but I don't know how to construct real gamma...

I’m New to the forum. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices. ##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...

I found in one book that every quadratic matrix 3x3 has at least one eigenvalue. I do not understand. Shouldn't be stated at least one real eigenvalue? Thanks for the answer.

Hello, I am working with heatmaps and similar. And I want to be able to add/subtract them and a few other custom operations. Here is a quick rundown: 1: Generate nxn (or nxm if possible) matrix, each element represents a heatmap color, or alternatively this could simply be defined as an image...

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}...

Well, the question is in the title.

It says in any textbook (for example, in classical text «Theory of matrices» by P. Lankaster) on matrix theory that matrices form an algebra with the following obvious operations: 1) matrix addition; 2) multiplication by the undelying field elements; 3) matrix multiplication. Is the last one...

I have a trouble showing proofs for matrix problems. I would like to know how A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix holds for square matrix A. It would be great if you can show how one leads to another with examples! :) Thanks for helping...

I got that ##{x_u}{y_v}-{x_y}{y_u}=####\frac{1}{\frac{1}{{x_u}{y_v}}-\frac{1}{{y_u}{x_v}}}##. But this implies that ##{x_u}{x_v}{y_u}{y_v}=-1## and I don't see how that is true?

Product of two Hermitian matrix ##A## and ##B## is Hermitian matrix only if matrices commute ##[A,B]=0##. If that is not a case matrix ##C=AB## could have complex eigenvalues. If A=\sum_k \lambda_k|k \rangle \langle k| B=\sum_l \lambda_l|l \rangle \langle l| AB=\sum_{k,l}\lambda_k\lambda_l|k...

The summary pretty much explains my question. I know that ##\left[ A, e^B \right]=0## if ##[A,B]=0## (and can prove it), but I can't figure out how to prove if it is or is not an "if and only if" statement. Thanks in advance!

I just realized quantum operators X and P aren't actually just generalizations of matrices in infinite dimensions that you can naively play with as if they're usual matrices. Then I learned that the space of quantum states is not actually a Hilbert space but a "rigged" Hilbert space. It all...

"Given a 3D Harmonic Oscillator under the effects of a field W, determine the matrix for W in the base given by the first excited level" So first of all we have to arrange W in terms of the creation and annihilation operator. So far so good, with the result: W = 2az2 - ax2 - ay2 + 2az+ 2 -ax+...

Summary

I assumed a column vector of degree 3 and then calculated A from the given condition.But after solving it i tried to find A2 and then I got stuck as none of the options seem to match. Please help. I think i will have to learn LATEX.🙁🙁

I am reading Kristopher Tapp's book: Matrix Groups for Undergraduates. I am currently focused on and studying Section 1 in Chapter2, namely: "1. Complex Matrices as Real Matrices". I need help in fully understanding how to prove an assertion related to Tapp's Proposition 2.4. Proposition 2.4...

At the start of Chapter 9, M. A. Armstrong in his book, "Groups and Symmetry" (see text below) writes the following: " ... ... Each matrix $$A$$ in this group determines an invertible linear transformation $$f_A: \mathbb{R} \to \mathbb{R}$$ defined by $$f_A(x) = x A^t$$ ... ... "I know that one...

$$c'\begin{pmatrix}u \\ v \end{pmatrix}=\begin{pmatrix}\frac{\partial f^1}{\partial u} & \frac{\partial f^1}{\partial v} \\\frac{\partial f^2}{\partial u} & \frac{\partial f^2}{\partial v}\end{pmatrix} $$ There is a problem with the first line of the matrix, but I am not too sure what it is.

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...

Summary:: Suppose that [x, y] = e^{-3t} [-2, -1] is a solution to the system $x' = Ax$, where A is a matrix with constant entries. Which of the following must be true? a. -3 is an eigenvalue of A. b. [4, 2] is an eigenvector of A. c. The trajectory of this solution in the phase plane with axes...

Let me start by saying that my question will be somewhat vague by mathematical standards. I'm not a mathematician! I'm looking for some intuition about how defective matrices are distributed in the space of all matrices. I understand that they are rare and in some sense discontinuous - matrices...

Motivation: Due to the spectral theorem a complex square matrix ##H\in \mathbb{C}^{n\times n}## is diagonalizable by a unitary matrix iff ##H## is normal (##H^\dagger H=HH^\dagger##). If H is Hermitian (##H^\dagger=H##) it follows that it is also normal and can hence be diagonalized by a...

Being myself a chemist, rather than a physicist or mathematician (and after consulting numerous sources which appear to me to skip over the detail): 1) It’s not clear to me how one can go generally from a choice of basis vectors in real space to a representation matrix for a spatial symmetry...

When working on the exercise 3.2 of Peskin's QFT, I find one of the calculating steps confused for me. I read the solution, which is showed in the picture. I just don't understand the boxed part. I know it involved the Dirac equation, and the solution seems to treat the momentum as a operator...

Hello, everyone. :) All I could gather is that, if I'm correct, lattices are spans of the column vectors of the matrix within the "LAT()" notation and the X and Y occurrences are unit placeholders (such as the pixel unit (since this is in the context of image processing)). And, as an attempt...

I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...

Suppose I have a vector of matrices: \mathbf{v}=(A_{1},\cdots,A_{n}) How would I vectorise this in MATLAB? This question comes from a requirement to compute a Greens function for the spherical heat equation. I can easily compute a single function for a single position in space, but can I do...

I have tried to do this using arrays and do loops: program matrixmul implicit none real A(2, 2), B (2, 2), C (2, 2) integer i, j, k write (*, *) 'Input: First matrix' do i = 1, 2 do j = 1, 2 read (*, *) A (i, j) enddo enddo write (*, *) 'Input: Second...

What are the rules to write Pauli's spin matrices in higher-order matrices (especially in 4x4 matrices)

what are Dirac's gamma matrices . especially , does it have many forms?

I first tried by assuming the matrices but it was becoming complicated so i tried taking transpose on both sides,it also did not help.So now i could not think of what to do further.Help please.

How do the Pauli spin matrices transform under an inversion ? I think I mean to say the 3 dimensional improper rotation which is just in 3 dimensional matrix notation minus the identity - so exactly how are the 2 dimensional Pauli spin matrices changed. And under a 180 rotation do the 'y' and...