Matrix Definition and 1000 Threads

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}...

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...

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 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...

I have a .txt file which is 50 rows x 50 columns filled with entirely 0's and 1's. I have tried to plot the data with and without spaces between each column. I keep getting this message: gnuplot> splot 'C:\Users\raf\Desktop\PolymerProject\monte carlo code\directionInitial.txt' with pm3d...

Hello I was learning about determinants and matrices. I learned the generalization of getting the determinant of an n by n matrix. I then applied this to vector space (i + j + k) via a cross product and noticed that you leave the i j and k in their own columns in the first row of the matrix...

I've been using rotation matrices for quite some time now without fully grasping them. Whenever I tried to develop an intuitive understanding of... x' = x\cos\theta - y\sin\theta \\ y' = x\sin\theta + y \cos\theta ... I failed and gave up. I've looked at numerous online texts and videos, but...

Homework Statement Consider the linear transformation T from V = P2 to W = P2 given by T(a0 + a1t + a2t2) = (−4a0 + 2a1 + 3a2) + (2a0 + 3a1 + 3a2)t + (−2a0 + 4a1 + 3a2)t^2 Let E = (e1, e2, e3) be the ordered basis in P2 given by e1(t) = 1, e2(t) = t, e3(t) = t^2 Find the coordinate matrix...

Homework Statement The weighted vector norm is defined as ##||x||_W = ||Wx||##. W is an invertible matrix. The induced weighted matrix norm is induced by the above vector norm and is written as: ##||A||_W = sup_{x\neq 0} \frac{||Ax||_W}{||x||_W}## A is a matrix. Need to show ##||A||_W =...

Homework Statement Let u1,u2,u3 be an orthonormal basis for R3 and consider M as the plane with equation x1+2x2-2x3=0. Find the matrix of orthogonal reflection in that plane with respect to the given basis. Homework EquationsThe Attempt at a Solution In previous exercises , I had a matrix A...

Dear all, I want to know how to convert operator matrix when using Dirac Bra-Ket notation when it must be converted into a new dimension. I am currently working on transition dipole moment operator matrix D which I am going to use the following one: D = er Where e is charge of electron, r is...

Homework Statement $$ \begin{bmatrix} a &b \\ c&d \end{bmatrix}$$ I'm supposed to find the inverseHomework Equations Method of Gauss-Jordan The Attempt at a Solution So I tried putting zeros in this and I got the following : $$ \begin{bmatrix} ad-ac &0 &ad &ad-a \\ 0&bc-ad &c &-a...

Homework Statement Show that all square matrix (A whatever) can be written as the sum of a symmetric matrix and a anti symmetric matrix. Homework Equations I think this relation might be relevant : $$ A=\frac{1}{2}*(A+A^{T})+\frac{1}{2}*(A-A^{T}) $$ The Attempt at a Solution I know that we...

Homework Statement Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...

Homework Statement Write in Vector-Matrix form then write the augmented matrix of the system. r + 2s + t = 1 r - 3s +3t = 1 4s - 5t = 3 Homework Equations The matrix to which the operations will be applied is called the augmented matrix of the system Ax = b, It is formed by appending the...

Homework Statement [/B] Find the limit as ##n \to \infty ## of ##U_n(a) =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & a/n \\ 0 & -a/n & 1 \end{pmatrix}^n##, for any real ##a##. Homework EquationsThe Attempt at a Solution I find ##U =\begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos a & \sin a \\ 0 & -\sin a &...

Homework Statement Prove $$||UA||_2 = ||AU||_2$$ where ##U## is a n-by-n unitary matrix and A is a n-by-m unitary matrix. Homework Equations For any matrix A, ##||A||_2 = \rho(A^*A)^.5##, ##\rho## is the spectral radius (maximum eigenvalue) where ##A^*## presents the complex conjugate of A. U...

Homework Statement If Γ is a k-regular simple graph and Γ its directed double, show that the matrix ˜ S for Γ (as per the FEATURED ARTICLE ) is a multiple of the adjacency matrix ˜ for Γ. Find the multiple. Assume k > 1. The matrix S is the probability matrix. The probability of going from one...

Homework Statement Find a 3x3 matrix A that satisfies the following equation where x, y, and z can be any numbers. ## A \begin{vmatrix} x \\ y \\ z \end{vmatrix} = \begin{vmatrix} x + y \\ x - y \\ 0 \end{vmatrix}##Homework EquationsThe Attempt at a Solution I attempted to solve this like...

So, I was examining the ground state of a Bose-Hubbard dimer in the negligible interaction limit, which essentially amounts to constructing and diagonalizing a two-site hopping matrix that has the form H_{i,i+1}^{(n)} = H_{i+1,i}^{(n)} = - \sqrt{i}\sqrt{n-i+1}, with all other elements zero...

Hi, I am looking for the general form of 2x2 complex transformation matrix. I have the article, that says "the relative position of a self-adjoint 2x2 matrix B with respect to A as a reference (corresponding to the transformation from the eigenspaces of A to the eigenspaces of B) is determined...

Hello, I am kind of new to Matlab so the questions I will ask probably sound a bit basic. Anyways, here goes: I want to create the matrix below which has both constants and variables. How can I do this? I know how to create a normal matrix (e.g. B = [1 0 2; 3 4 5; 0 2 3]) but I don't know how...

What does it mean by "In the position representation -- in which r is diagonal" in the paragraph below? How can we show that? Does it mean equation (3) in http://scienceworld.wolfram.com/physics/PositionOperator.html? (where I believe the matrix is in the ##|E_n>## basis)

Hello fellow nerds, I've come across a math problem, where I'd like to find the solution vector of a NxN square matrix. It is possible to find a solution for a given N, albeit numbers in the matrix become very large for any N>>1, and numbers in the solution vector become very small. So it's not...

I want to construct a completely correlated chi^2. I have a two-dimensional dataset, and its basically like: {m1,m2,m3,m4} {a1,a2,a3,a4} {x0,x0,x0,x0} So m1-m4, a1-a4 are all different, but each x0 is the same. This happens when I am fitting 2D data, but it is required that the function goes...

Hi everyone, I need help for preparing a Hamiltonian matrix. What will be the elements of the hamiltonian matrix of the following Schrodinger equation (for two electrons in a 1D infinite well): -\frac{ħ^{2}}{2m}(\frac{d^{2}ψ(x_1,x_2)}{dx_1^{2}}+\frac{d^{2}ψ(x_1,x_2)}{dx_2^{2}}) +...

The induced matrix norm for a square matrix ##A## is defined as: ##\lVert A \rVert= sup\frac{\lVert Ax \rVert}{\lVert x \rVert}## where ##\lVert x \rVert## is a vector norm. sup = supremum My question is: is the numerator ##\lVert Ax \rVert## a vector norm?

Homework Statement I can't understand this paper. I understand the whole incidence matrix stuff, but I don't quiet get how it relates to the linear algebra. I don't know if this is allowed to do, but I will ask you questions line by line, so basically you will read the paper with me explaining...