Matrix Definition and 1000 Threads

Homework Statement Nearly free electron model in a 2D lattice. Consider a divalent 2D metal with a square lattice and one atom per primitive lattice cell. The periodic potential has two Fourier components V10 and V11, corresponding to G = (1,0) and (1,1). Both are negative and mod(V10) >...

This is probably a simple question A factory produce a good that requires 3 labor-hours in the assembly department and 1 labor-hour in the finishing department. Assembly personnel receive 19 per hour and finishing personnel receive 16 per hour. I need to write a matrix or vector product that...

'Homework Statement Find the matrix A' for T: R2-->R2, where T(x1, x2) = (2x1 - 2x2, -x1 + 3x2), relative to the basis B' {(1, 0), (1, 1)}. Homework Equations B' = {(1, 0), (1, 0)} so B'-1 = {(1, -1), (0, 1)}. The Attempt at a Solution I'm confused at what exactly a transform matrix...

In special relativity, the electromagnetic field is represented by the tensor $$F^{\mu\nu} = \begin{pmatrix}0 & -E_{x} & -E_{y} & -E_{z}\\ E_{x} & 0 & -B_{z} & B_{y}\\ E_{y} & B_{z} & 0 & -B_{x}\\ E_{z} & -B_{y} & B_{x} & 0 \end{pmatrix}$$ which is an anti-symmetric matrix. Recalling the...

Hi! As the title says, what is the derivative of a matrix transpose? I am attempting to take the derivative of \dot{q} and \dot{p} with respect to p and q (on each one). Any advice?

It says in Susskind's TM: ##\langle L \rangle = Tr \; \rho L = \sum_{a,a'}L_{a',a} \rho_{a,a'}## with ##a## the index of a basisvector, ##L## an observable and ##\rho## a density matrix. Is this correct? What about the trace in the third part of this equation?

Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}.Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...

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...

Am I on the right path here? 1. Homework Statement i. Prove that ##T_{a}## and ##T_{b}## are linear transformations. ii. Compose the two linear transformations and show the matrix that represents that composition. 2. The attempt at a solution i. Prove that ##T_{a}## and ##T_{b}## are linear...

Let A = (a_{ij}) be a k\times n matrix of rank k . The k row vectors, a_i are linearly independent and span a k-dimensional plane in \mathbb{R}^n . In "Geometry, Topology, and Physics" (Ex 5.5 about the Grassmann manifold), the author states that for a matrix g\in...

Homework Statement A system is subject to Hamiltonian ##\hat{H}=-\gamma B_z \hat{S}_z##. Write down the density matrix.[/B]Homework Equations For canonical ensemble ##\hat{\rho}=\frac{1}{Tr(e^{-\beta \hat{H}})}e^{-\beta \hat{H}}## In general ##\rho=\sum_m |\psi_m\rangle \langle \psi_m|## The...

i3 -? This is how I tried to solve using cramer's rule (denominator is the determinant of the matrix, while numerator is the determinant of the modified matrix) I ignored the prefix Kilo, but the method is right. So I get a different value for i3. Does anyone know what's wrong?

This is where I am stuck. I studied ordered basis and coordinates vector previous to this. of course I studied vector space, basis, linear... etc too, However I can't understand just this part. (maybe this whole part) Especially this one which says [[T(b1)]]c...[[T(bn)]]c be a columns of...

I have created a matrix with a class called Lattice. The lattice is filled with objects of type 'Dipole' which is created with another class. The problem I am having is with boundary conditions when I look for a neighbour. e.g If i pick a dipole on the top row, I want its 'above' neighbour to be...

Homework Statement Imagine we have a tensor ##X^{\mu\nu}## and a vector ##V^{\mu}##, with components ## X^{\mu\nu}=\left( \begin{array}{cccc} 2 & 0 & 1 & -1 \\ -1 & 0 & 3 & 2 \\ -1 & 1 & 0 & 0 \\ -2 & 1 & 1 & -2 \end{array} \right), \qquad V^{\mu} = (-1,2,0,-2). ## Find the components of...

Hi, suppose two players are a playing a game with a non-square payoff matrix, like for example this one: ...a...b... A: (1,3) (1,0 B: (0,0) (2,1) C: (3,1) (0,3) How would one go about finding an optimal mixed strategy for something like this? I mean, if this was a 3x3 matrix then one could find...

Homework Statement Let A be an n×n matrix which has the property that A^2 =A. (i) Write down the most general polynomial in AHomework EquationsThe Attempt at a Solution My biggest problem is that I don't even understand what the question is asking Is it just sum (alphaA^n)=0 but A^n=A...

Here is a link the code which I am trying to modify. It is the radial average of a matrix function. http://uk.mathworks.com/matlabcentral/fileexchange/46468-radialavg-zip/content/radialavg.m I want to restrict the function to only look within certain angles, e.g. 15 degrees either side of the...

How would you go about doing this, I see it so often quoted as a method, but no-where can I find an example This is what I was thinking D=P^(-1)AP Would it then follow that D^(-1)=P^(-1)A^(-1)P ? My reasoning being: DD^(-1)= P^(-1)APP^(-1)A^(-1)P identity matrix= P^(-1)AA^(-1)P=identity...

Homework Statement I'm told to find the matrix Q of the matrix A Homework EquationsThe Attempt at a Solution So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3 S = [X_1, X_2, X_3] but...

Homework Statement This is actually part 3 of the question. Part one was to form 7 equations to form a 7x7 matrix, part 2 was to solve it, which I've done. This question is to be done with Matlab, by the way. Part 3: Homework Equations Frequency response = Vout / Vin. The Attempt at a...

Homework Statement For problems 1 and 2 use http://T: R^3 to R^3, T(<x1,x2,x3>) = <2x1-x2, x2+3x3, x1 - x2+2x3>, , T: R^3 to R^3, T(<x1,x2,x3>) = <2x1-x2, x2+3x3, x1 - x2+2x3>, , bases B = { <1,0,1>, <1,1,0>, <0,1,1> } and B' = { <1,1,-2>, <2,1,-1>, <3,1,1> }. Find T ( <3,-1,2> ) by using...

Homework Statement A = 000 010 101 Find Eigenvalues, its corresponding eigenvectors, and find a matrix W such that W^t*AW = D, where D is a diagnol matrix.(note that W^t represents the transpose of W) Homework Equations Eigenvalues, Eigenvectors, diagnolization[/B]The Attempt at a...

Homework Statement If A100 is some 3x3 matrix, find the base matrix A. 2. Relevant information Eigenvalues, diagonalization, etc. The Attempt at a Solution So far, I've been finding the eigenvalues and diagonalizing the matrix via A = P-1DP where D is the diagonal matrix and P is a matrix...

Homework Statement I feel like I should know the answer to this, so I believe this to be an easy question. Say have matrix A, and I store the elimination matrices E_1,1 E_2,1 etc. and somewhere in the elimination process I have to use a permutation matrix to swap rows. My question is when I...

Homework Statement $$\begin{bmatrix} a_{11} & a_{12} & 0 & 0\\ a_{12} & a_{22} & a_{23} & 0\\ 0 & a_{23} & a_{33} & a_{34} \\ 0 & 0 & a_{34} & a_{44} \\ \end{bmatrix} = \begin{bmatrix} q_{11} & q_{12} & q_{13} & q_{14} \\ q_{21} & q_{22} & q_{23} & q_{24} \\ q_{31} & q_{32} & q_{33} & q_{34}...

Homework Statement Suppose the vector ##\phi## transforms under SU(2) as: $$\phi' = (\exp(-i \alpha \cdot t))_{ij}\phi_j,$$ where ## (t_j)_{kl} = −i \epsilon_{jkl}## and ##j, k, l \in \left\{1, 2, 3\right\}.## Based on ##\phi,## we define the ##2 \times 2## matrix ##\sigma = \tau \cdot...

Is the Hamiltonian matrix that is constructed in Ch 8 of the Feynman lectures the Heisenberg matrix picture, or is it something else? I am just curious. Thanks, Chris Maness

I want to rotate an inclined plane to achieve a flat surface. I think I can use the Euler angles to perform this operation. Using following data: and following rotation matrix I think you can make the plane flat by following rotations: 1: rotation around x-axis by 45° 2: rotation around...

Hi all, I have this data that can be described by M*A = B, where M is a 3x3 matrix and A and B are 3x1 vectors. Since I know and can collect A and B data, and I have 9 unknowns in the 3x3 matrix, I thought that collecting 9 pairs of A and B vectors would yield the matrix M's coefficients via 9...

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##...

I still can't find a book with properties and theorems involving block matrices multiplication to reference in my undergraduate work. thanks

Homework Statement Homework EquationsThe Attempt at a Solution The answer in the solutions is given as : (2x+1)(x-1)(1-x), they did their matrix differently so that's how they got that answer. I used wolfram alpha to factorise my quadratic on the last line and it gave me alternative forms...

I know row reduction methods are the best way to calculate the determinant of large matrices. I was wondering if you can use the appended matrix method to calculate the determinant of a 4x4 by appending the matrix with the first 3 columns. There should be n! terms, but I only get 8 instead of 24.

Hi to all. Say that you have an eigenvalue problem of a Hermitian matrix ##A## and want (for many reasons) to calculate the eigenvalues and eigenstates for many cases where only the diagonal elements are changed in each case. Say the common eigenvalue problem is ##Ax=λx##. The ##A## matrix is...

Homework Statement Let A be an n x n matrix, and let v, w ∈ ℂn. Prove that Av ⋅ w = v ⋅ A†w Homework Equations † = conjugate transpose ⋅ = dot product * = conjugate T = transpose (AB)-1 = B-1A-1 (AB)-1 = BTAT (AB)* = A*B* A† = (AT)* Definitions of Unitary and Hermitian Matrices Complex Mod...

Hi, I'm actually trying to do some simulation on matrix converters for my university power electronics subject. I am reading through the different types of modulation techniques. One of them, the Venturini direct approach, with sinusoidal voltage output and input current. I'm having a little...

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...

Homework Statement Show that there is a non-singular matrix M such that ##MAM^T = F## for any antisymmetric matrix A where the normal form F is a matrix with 2x2 blocks on its principal diagonal which are either zero or $$\begin{pmatrix} 0 &1 \\ -1&0 \end{pmatrix}$$ To do so, consider the...

Homework Statement I did poorly on my exam, which I thought was very fair, and am now trying to understand certain aspects of perturbation theory. There are a total of three, semi related problems which i have questions about. They are mainly qualitative in nature and involve an intuitive...

Ok, I've got these functions to get the x (right), y (up) and z (forward) coordinates to plot with my computer program: x = r*Math.cos(a)*Math.sin(o) y = r*Math.sin(a) z = -r*Math.cos(a)*Math.cos(o) It's the equations of a sphere where I've placed the origin (o,a,r) = (0,0,0) of the source...

Hi I'm just having trouble wrapping my head around differential equations with matrices and vectors... For example: let y be a vector. let A(t) be an nxn matrix. I have the differential equation: dy/dt = A(t)y So I think I understand why the solution is y = ceA(t) But I'm having trouble...

Homework Statement The assignment is to find all values of k (in R) for which the system has 0 solutions, 1 solution and infinite solutions. If there are infinite solutions, find the amount of free variables. The system of linear equations: kx + (k+1)y + z = 0 kx + y + (k+1)z = 0 2kx + y + z =...

Homework Statement I am designing a MIMO communication system, with input signal s, channel H and transform matrix T. The received signal is corrupted by noise. Homework Equations [/B] The received signal is r = Hs+n And then it is transformed (compressed) by: y = Tr And then its...

Homework Statement Homework Equations Inverse of an (nxn) (n=2 only) square matrix: The Attempt at a Solution The answer provided in the solutions does the exact same thing except, where my ?? are. It does A = BCB^-1. Where as I do A = CBB^-1. When I was doing this question I was...

1. Show that the map $\mathcal{A}$ from $\mathbb{R}^3$ to $\mathbb{R}^3$ defined by $\mathcal{A}(x,y,z) = (x+y, x-y, z)$ is a linear transformation. Find its matrix in standard basis. 2. Find the dimensions of $\text{Im}(\mathcal{A})$ and $\text{Ker}(\mathcal{A})$, and find their basis for the...

Homework Statement I'm struggling to find a solution to exercise (*b). I have uploaded a pdf of the assignment. Please advise me at your convenience. Homework Equations x(x_l^+) = T(x_l^+, x_l^-)x(x_l^-) The Attempt at a Solution x(a^-) = \frac{\psi(a^-)}{\psi(a^-)} , T(a^+, a^-) \left(...

Homework Statement I have to find the inverse for this generic matrix (the dimensions are not specified, but I assume its a square matrix, I don't know if that is necessary). ##A=\left [ \begin{matrix} 1 & -1 & -1 & -1 & \dots & -1 & -1 \\ 0 & 1 & -1 & -1 & \dots & -1 & -1 \\ 0 & 0 & 1 & -1...

Homework Statement I would like to know how to derive the quantum commutation relations in matrix form, $$i \hbar \partial_t x(t)= [x(t),E]$$ $$i \hbar \partial_t P(t)= [P(t),E]$$ Where X(t), P(t) and E are the position, momentum and the energy of the particle, respectively. 2. Homework...

Homework Statement Well, basically my issue isn't exactly with how to multiply matrices. I know how to do that, my issue is the way my lecturer shows how to find the size of the new matrix, this is all that he shows: I mean how is he getting AX to be a 3x1 matrix? Homework EquationsThe...