Matrices Definition and 1000 Threads

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...

https://www.physicsforums.com/attachments/8962 ok this is my overleaf homework page but did not do (c) and (d) this class is over but trying to do some stuff I missed. I am only auditing so I may sit in again next year...;) also if you see typos much grateful I don't see a lot of replies on...

Hi :) I have several questions about the Pauli matrices, I have seen them when the lecturer showed us Stern-Gerlach experiment , and we did some really weird assumptions on what we think they should be. 1- why did we assume that all of those matrices should satisfy σ2 = I (the identity...

Hi All, While trying to read a matrix from data file using fortran90 code ,I get garbage values and a backtrace error. Error termination. Backtrace: #0 0x7f4a4de3631a #1 0x7f4a4de36ec5 #2 0x7f4a4de3768d #3 0x7f4a4dfa4d42 #4 0x7f4a4dfa6ad5 #5 0x7f4a4dfa80f9 #6 0x56040bbeae57 #7...

he is asking for the division of the two matrices , so i tried to get the inverse of the matrix A but it appears to get more complex as the delta for A is somehow a big equation . and what really bothers me that there is another A , B inside the matrix B ?! find B/A .

My matrix text files can vary from 1x1 to 10x10 (Another file will be given when the code gets tested but it's all square matrices) I'm stuck here. #include <iostream> #include <fstream> #include <string> using namespace std; int main() { const int MAXI = 10; int x, y, z...

Suppose p(λ)=(λ-1)^3 for some diagonalizable matrix A. Calculate A^25. I'm confused as to how to approach this question without A being given. I thought perhaps I could use the characteristic equation in some way although I am still unsure. I think I could start with using λ=1. Would my...

Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...

Let us for a moment look a field transformations of the type $$\phi(x)\longmapsto \exp\left(\frac{1}{2}\omega_{\mu\nu}S^{\mu\nu}\right)\phi(x),$$ where ##\omega## is anti-symmetric and ##S^{\mu\nu}## satisfy the commutation relations of the Lorentz group, namely $$\left[S_{\mu \nu}, S_{\rho...

<Moderator's note: Moved from a technical forum and thus no template.> I am at the beginners level of linear algebra and having problem of the intersection of matrices. Your kind help is much appreciated for the following question Let\quad M1=\begin{Bmatrix} x & -x \\ y & z \end{Bmatrix},\quad...

I have equation system: x + y + z - a*k = 0 -b*x + y + z = 0 -c*y + z = 0 -d*x + y = 0 where: a, b, c, d = const. Have to find: x, y, z, k Attempt of solution: I create Matrix A with coefficients; Matrix B - Solutions (Zeros) and Matrix X - variables. When I try to use Cramer's rule -...

I am reading David Poole's book: "Linear Algebra: A Modern Introduction" (Third Edition) and am currently focused on Section 6.6: The Matrix of a Linear Transformation ... ... I need some help in order to fully understand Example 6.76 ... ... Example 6.76 reads as follows: My question or...

Hi,how do I go about answering the attached question? I know that for a matrix to have no solution, there needs to be a contradiction in some row. Unique solutions is when m* ${x}_{3}$ =c , where m* ${x}_{3}$ $\ne$ 0. One way I tried was if a=0, then from row (1) : b* ${x}_{3}$ =2...

Hi, I've a question that asks me to determine matrix A , where A= ${S}^{-1}$* B* S They have given matrices S and B in the question. I think the answer is A=B, since S * ${S}^{-1}$ would give me the identity matrix and anything multiplied by the identity matrix is itself. Is this correct?

Hi PF! Let's say we have a matrix that looks like $$ A = \begin{bmatrix} 1-x & 1+x \\ i & 1 \end{bmatrix} \implies\\ \det(A) = (1-x) -i(1+x). $$ I want ##A## to be singular, so ##\det(A) = 0##. Is this impossible?

Homework Statement The problem is attached. I'm working on the second system with the masses on a linear spring (not the first system). I think I solved part (a), but I'm not sure if I did what it was asking for. I'm not sure exactly what the question means by the... L=.5Tnn-.5Vnn. Namely, I'm...

Homework Statement Not for homework, but just for understanding. So we know that if a matrix (M) is orthogonal, then its transpose is its inverse. Using that knowledge for a diagonalised matrix (with eigenvalues), its column vectors are all mutually orthogonal and thus you would assume that...

Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...

For example, does this always hold true? M_ab = v_a × w_b If not, where does it break down?

Let's say we have a given matrix ##G##. I want to find a set of ##M## matrices so that ##MG = GM## and prove that this is a group. How can I approach this problem?

Hello! (Wave) If the matrix $A \in M_n(\mathbb{C})$ has $m_A(x)=(x^2+1)(x^2-1)$ as its minimal polynomial, then I want to find the minimal polynomials of the matrices $A^2$ and $A^3$. ($M_n(k)$=the $n \times n$ matrices with elements over the field $k=\mathbb{R}$ or $k=\mathbb{C}$) Is there a...

Homework Statement Homework Equations 3. The Attempt at a Solution [/B] ## |3 \rangle = |1 \rangle - 2 ~ |2 \rangle ## So, they are not linearly independent. One way to find the coefficients is : ## |3 \rangle = a~ |1 \rangle +b~ |2 \rangle ## ...(1) And solve (1) to get the values of a...

Hello, I'm having a visualisation problem. I have a point in R3 that I want to rotate about the ##y##-axis anticlockwise (assuming a right-handed cartesian coordinate system.) I know that the change to the point's ##x## and ##z## coordinates can be described as follows: $$z =...

Homework Statement Let ##n \in \mathbb{Z}^+## and let ##F## be a field. Prove that the set ##H = \{(A_{ij}) \in GL_n (F) ~ | ~ A_{ij} = 0 ~ \forall i > j \}## is a subgroup of ##GL_n (F)## Homework EquationsThe Attempt at a Solution So clearly the set is nonempty since ##I_n## is upper...

Hi. I have a question about covariance matrices (CMs) and a standard form. In Ref. [Inseparability Criterion for Continuous Variable Systems], it is mentioned that CMs ##M## for two-mode Gaussian states can be symplectic transformed to the standard form ##M_s##: ## M= \left[ \begin{array}{cc}...

$$\tiny{la1.1.22}$$ $\textsf{Construct 3 different augmented matrices for linear systems whose solution set is}$ $$x_1=3,\quad x_2=-2,\quad x_3=-1$$Well we could start just by$$3x-2y+y=-1$$but then we need a $3\times4$ matrix

Hello, I am having trouble comprehending how grids are made and defined in computers. What is the unit that they use and how is it defined ? I know that softwares use standardized units of measure (measurement) such as centimetre. Basically, how is a 3-Dimensional Space created in computers...

Hi PF! I am trying to solve the eigenvalue problem ##A v = \lambda B v##. I thought I'd solve this by $$A v = \lambda B v \implies\\ B^{-1} A v = \lambda v\implies\\ (B^{-1} A - \lambda I) v = 0 $$ and then using the built in function Eigenvalues and Eigenvectors on the matrix ##B^{-1}A##. But...

While writing down the basis for SU(2), physicists often choose traceless hermitian matrices as such, often the Pauli matrices. Why is this? In particular why traceless, and why hermitian?

Homework Statement Homework EquationsThe Attempt at a Solution I know I would have to do something with my calculator and I tried to solve like solving an equation for C, but not sure. I put all the matrices in my calculator. I then subtracted the first matrix to the other side then...

Prove that the maximum number of mutually commuting linearly independent complex matrices of order $n$ is equal to $\lfloor n^2/4\rfloor + 1.$

For finite matrices ##A## and ##B## we have Tr(AB)=Tr(BA) What happens in case of infinite matrices?

Suppose I have a hermitian ##N \times N## matrix ##M##. Let ##U \in SU(N)## be the matrix that diagonalizes ##M##: ##M = U\Lambda U^\dagger##, where ##\Lambda## is the matrix of eigenvalues of ##M##. This transformation can be considered as the adjoint action ##Ad## of ##SU(N)## over its...

Matrix \left[ \begin{array}{rr} 1 & 1 \\ 0& 0 \\ \end{array} \right] is not symmetric. When we find eigenvalues of that matrix we get ##\lambda_1=1##, ##\lambda_2=0##, or we get matrix \left[ \begin{array}{rr} 1 & 0 \\ 0& 0 \\ \end{array} \right]. First matrix is not hermitian, whereas second...

Suppose I have a matrix M = A + εB, where ε << 1. If A is invertible, under some assumptions I can write e Neumann series M-1 = (I - εA-1B)A-1 But if A is not invertible, how can I expand M-1 in powers of ε? Thanks in advance

Homework Statement I need some help with a question on my assignment. It asks to set up a matrix from the linear equations, y=25x+70 and y=35x+40. Homework Equations How do I set this matrix up? The Attempt at a Solution I think that I have to rewrite it as 25x-y=-70 and 35x-y=-40. But then I...

Homework Statement I'm trying to show the Eigenstate of S2 is 2ħ^2 given the matrix representations for Sx, Sy and Sz for a spin 1 particle Homework Equations Sx = ħ/√2 * \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} Sy = ħ/√2 * \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i...

Homework Statement Show that for a second order cartesian tensor A, assumed invertible and dependent on t, the following holds: ## \frac{d}{dt} det(A) = det(a) Tr(A^{-1}\frac{dA}{dt}) ## Homework Equations ## det(a) = \frac{1}{6} \epsilon_{ijk} \epsilon_{lmn} A_{il}A_{jm}A_{kn} ## The...

So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...

Is there a notion of “coherent” operations on Jacobian matrices? By this I mean, an operation on a Jacobian matrix A that yields a new matrix A' that is itself a Jacobian matrix of some (other) system of functions. You can ascertain whether A' is coherent by integrating its partials of one...

Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. Homework Equations I know the 8 axioms about the vector spaces. The...

Homework Statement Matrices ##\alpha_k=\gamma^0 \gamma^k##, ##\beta=\gamma^0## and ##\alpha_5=\alpha_1\alpha_2\alpha_3 \beta##. If we know that for Dirac Hamiltonian H_D\psi(x)=E \psi(x) then show that \alpha_5 \psi(x)=-E \psi(x) Homework EquationsThe Attempt at a Solution I tried to...

"It is easily shown" that the gamma 5 matrix anticommutes with the four gamma matrices. Can someone tell me how or provide a link to such proof?

Hi, I just have a question relative to matrices, mostly. Is the reason there are 4 values in a matrix because there are (at least in basic terms) 3 dimensions of space and one of time? Like it seems kind of obvious, but for some weird reason in school they never state it explicitly in those...

Homework Statement Given the six vectors below: 1. Find the largest number of linearly independent vectors among these. Be sure to carefully describe how you would go about doing so before you start the computation. 2 .Let the 6 vectors form the columns of a matrix A. Find the dimension of...

Hi, the three main types of complex matrices are: 1. Hermitian, with only real eigenvalues 2. Skew-Hermitian , with only imaginary eigenvalues 3. Unitary, with only complex conjugates. Shouldn't there be a fourth type: 4. Non-unitary-non-hermitian, with one imaginary value (i.e. 3i) and a...

Homework Statement Construct a 3 × 3 example of a linear system that has 9 different coefficients on the left hand side but rows 2 and 3 become zero in elimination. If the right hand sude of your system is <b1,b2,b3> (Imagine this is a column vector) then how many solutions does your system...

Homework Statement (i) Reduce the system to echelon form C|d (ii) For k = -12, what are the ranks of C and C|d? Find the solution in vector form if the system is consistent. (iii) Repeat part (b) above for k = −18 Homework Equations Gaussian elimination I used here...

I understand that momentum, rest mass and energy can be put on the sides of a right triangle such that the Pythagorean Theorem suggests E^2=p^2+m^2. I understand that the Dirac equation says E=aypy+axpx+azpz+Bm and that when we square both sides the momentum and mass terms square while the cross...

Homework Statement The object is made out of multiple parts. The inertia matrices of every part are given. Only one part is rotating. How do I find the total inertia matrix. Homework EquationsThe Attempt at a Solution I thought that I could sum the inertia matrices, after tranforming them to...