Matrix Definition and 1000 Threads

I need to find a matrix representation for operator A=x\frac{d}{dx} using Legendre polinomials as base. I would use a_{mn}=\int^{-1}_{-1}P_m(x)\,x\frac{d}{dx}\,P_n(x)\,dx, but I have the problem that Legendre polinomials aren't orthonormal \langle P_{i}|P_{l}\rangle=\delta_{il}\frac{2}{2i+1}. I...

It is possible to find area of triangle or parallelogram in euclidean by using matrix determinant composed of unity, x coeffs and y coeffs in row1,2,3 respectively. Is it possible to do that in higher dimensions as well although it may be not as simple as in 2D case. In 3d matrix composed of...

Many important techniques in fields such as CT and MR imaging in medicine, nondestructive testing and scientific visualization are based on trying to recover a matrix from its projections. A small version of the problem is given the sums of the rows and columns of a 2 x 2 matrix, determine the...

Homework Statement Let ##B \in M_n (\mathbb{C})## be such that ##B \ge 0## (i.e., it is a positive semi-definite matrix) and ##b_{ii} = 1## (ones along the diagonal). Show that there exists a collection of ##n## unit vectors ##\{e_1,...,e_n \} \subset \mathbb{C}^n## such that ##b_{ij} = \langle...

MIT OCW 18.06 Intro to Linear Algebra 4th edt Gilbert Strang Ch6.2 - the textbook emphasized that "matrices that have repeated eigenvalues are not diagonalizable". imgur: http://i.imgur.com/Q4pbi33.jpg and imgur: http://i.imgur.com/RSOmS2o.jpg Upon rereading...I do see the possibility...

Homework Statement Let |v> ∈ ℂ^2 and |w> = A|v> where A is an nxn unitary matrix. Show that <v|v> = <w|w>. Homework Equations * = complex conjugate † = hermitian conjugate The Attempt at a Solution Start: <v|v> = <w|w> Use definition of w <v|v>=<A|v>A|v>> Here's the interesting part Using...

Homework Statement Let A∈M2x2(ℝ) such that ATA = I and det(A) = -1. Prove that for ANY such matrix there exists an angle θ such that A = ## \left( \begin{array}{cc} cos(\theta) & sin(\theta)\\ sin(\theta) & -cos(\theta)\\ \end{array} \right) ## It is not sufficient to show that this matrix...

Is any of the conjectures in: http://arxiv.org/pdf/hep-th/9610043v3.pdf have been proven/disproven? what has been left still open? I am thinking of reading this article sometime in the future, hope it's digestable (but first need to finish my studies of QFT and GR.)

Hello! (Wave) I have written the following code in matlab: function v=uexact(x,t) v=sin(2*pi*x)*exp(-4*pi^2*t); end function [ex]=test3 h = 1/50; T=1/2500; x=0:h:1; t=0:T:1; ex=uexact(x,t); end I...

I have just learned the first order system of ODE, i found that the Wronskian in second order ODE is |y1 y2 ; y1' y2'| but in first order system of ODE is the Wronskian is W(two solution), i wonder which ones is the general form? thank you very much

I haven't taken a course on quantum mechanics yet, but I was asked to solve (numerically) ##[-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)]\phi(x)=E\phi(x) ## ##V(x)=2000(x-0.5)^2## by supposing the solution is ##\sum_{0}^{\infty} a_n \phi_n(x)## and ##\phi_n(x)## is the typical solution to the a...

I have had no problem while finding the eigen vectors for the x and y components of pauli matrix. However, while solving for the z- component, I got stuck. The eigen values are 1 and -1. While solving for the eigen vector corresponding to the eigen value 1 using (\sigma _z-\lambda I)X=0, I got...

Please help me understand the following step

Homework Statement Show that no matrix A ∈ M3 (ℝ) exists so that A2 = -I3 Homework EquationsThe Attempt at a Solution This is from a french textbook of first year linear algebra. I'm quite familiar with properties of matrices but I don't have any idea of how to prove this. Thanks for the help!

Homework Statement Let L be a mapping such that L(A) = A^t, the transpose mapping. Find a matrix representing L with respect to the standard basis [1,1,1,1] Homework EquationsThe Attempt at a Solution So should I end up getting a 4x4 matrix here? I got 1,0,0,0 for the first column, 0,0,1,0 for...

Homework Statement A thin lens is placed 2m after the beam waist. The lens has f = 200mm. Find the appropriate system matrix. This is a past exam question I want to check I got right. Homework Equations For some straight section [[1 , d],[0 , 1]] and for a thin lens [[1 , 0],[-1/f , 1]]...

Homework Statement Is the moment of inertia matrix a tensor? Hint: the dyadic product of two vectors transforms according to the rule for second order tensors. I is the inertia matrix L is the angular momentum \omega is the angular velocity Homework Equations The transformation rule for a...

Hello everyone, I'm currently building the covariance matrix of a large dataset in order to calculate the Chi-Squared. The covariance matrix has this form: \begin{bmatrix} \sigma^2_{1, stat} + \sigma^2_{1, syst} & \rho_{12} \sigma_{1,syst} \sigma_{2, syst} & ... \\ \rho_{12} \sigma_{1,syst}...

Homework Statement Page 133 Homework Equations n/a The Attempt at a Solution What is the process for rewriting the third column? 2x-3 and be rewritten as 2x, and 2-3cosx can be rewritten as 2. I don't get this.

Hello I am trying to learn linear algebra, and I came across this definition of basis minor on this webpage: https://en.wikibooks.org/wiki/Linear_Algebra/Linear_Dependence_of_Columns "The rank of a matrix is the maximum order of a minor that does not equal 0. The minor of a matrix with the...

Homework Statement Demonstrate that matrix ##T## represents a 2nd order tensor ##T = \pmatrix{ x_2^2 && -x_1x_2 \\ -x_1x_2 && x_1^2}## Homework Equations To show that something is a tensor, it must transform by ##T_{ij}' = L_{il}L_{jm}T_{lm}##. I cannot find a neat general form for ##T_{ij}##...

Homework Statement Shown In the picture. I went to the prof for help he said and i quote :" don't use laplas expansion to find the determinate, it will take you for ever." Homework Equations I don't even know how to do this. prof had no notes on this and Boas is a god awful book for learning...

Homework Statement Hi! I have the 3x3 matrix for L below, which I calculated. But now I need to figure out how the equation below actually means! Is it just the inverse of L (L^-1)? I cannot proceed if I don't know this step. Homework Equations See image The Attempt at a Solution I put in...

Hello! (Wave) Suppose that we are given this $(N+1) \times (N+1)$ matrix: $\begin{bmatrix} -(1+h+\frac{h^2}{2}q(x_0)) & 1 & 0 & 0 & \dots & \dots & 0 \\ -1 & 2+h^2q(x_1) & -1 & 0 & \dots& \dots & 0\\ 0 & -1 & 2+h^2q(x_2) & -1 & 0 & \dots & 0\\ & & & & & & \\ & & & & & & \\ & & & & & & \\...

Homework Statement Compute ##A^j~\text{for} ~~j=1,2,...,n## for the block diagonal matrix##A=\begin{bmatrix} J_2(1)& \\ &J_3(0) \end{bmatrix}##, And show that the difference equation ##x_{j+1}=Ax_{j}## has a solution satisfying ##|x_{j}|\rightarrow\infty~\text{as}~j\rightarrow\infty##...

Hey guys, So I'm stuck on another question from the previous one that I posted and would absolutely love it if I can get some help regarding how to attempt this. I literally have no clue at how to go by solving it. I have a feeling for question one that the cosine laws might come in handy but...

Hello, I couldn't give the full explanation in the title - I am talking about a particular matrix. Given the matrix: A[1] = 0 0 1 1 0 0 1 1 0 A[5] = 1 1 1 1 1 1 1 1 1 Once it gets to the 5th boolean power, it becomes all 1's, and any power greater than or equal to 5 will always produce a...

Homework Statement "Calculate the max, min, count, average, and standard deviation (std dev) of a set of numbers. The formula for average is: average is sum divided by count The formula for standard deviation is stddev is the square root of the variance The formula for variance is variance is...

I'm trying to understand the maths of QM from Shankar's book - Principles of Quantum Mechanics: On page 21 of that book, there is a general derivation that if we have a relation: |v'> = Ω|v> Where Ω is a operator on |v> transfroming it into |v'>, then the matrix entries of the operator can be...

Homework Statement find eigenvalues and eigenvectors for the following matrix |a 1 0| |1 a 1| |0 1 a| Homework EquationsThe Attempt at a Solution I'm trying to find eigenvalues, in doing so I've come to a dead end at 1 + (a^3 - lambda a^2 -2a^2 lambda + 2a lambda^2 + lambda^2 a - lambda^3 - a...

I'd like to prove the fact that - since a rotation of axes is a length-preserving transformation, the rotation matrix must be orthogonal. By the way, the converse of the statement is true also. Meaning, if a transformation is orthogonal, it must be length preserving, and I have been able to...

Quick question, not even sure if I should post it here, but I can't think where else. If I wanted to write the short hand of A is an invertible matrix, would it be ok to just write A \ \exists A^{-1} ?

Homework Statement The moment of the couple is 600k (N-m). What is the angle A? F = 100N located at (5,0)m and pointed in the positive x and positive y direction -F = 100N located at (0,4)m and pointed in the negative x and negative y direction Homework Equations M = rxF M = DThe Attempt at a...

We want to know the square and the cube of each of the following numbers: 1, 2 and 3. You can have MATLAB obtain these results as follows. Create a vector [1 2 3] . Create a vector [2 3]. Use these two vectors as inputs to the meshgrid function (with the 3-element vector as the first argument)...

Please I need your help in such problems.. in terms of ladder operators to simplify the calculation of matrix elements... calculate those i) <u+2|P2|u> ii) <u+1| X3|u> If u is different in both sides, then the value is 0? is it right it is 0 fir both i and ii? when exactly equals 0, please...

Hi, I'm new in physics and optics so I need a little help. I've a simple optical system from 2 thin lenses. The first thin lens has a focal distance of 50 [mm] , and the second one has 25 [mm]. The 2 lenses are separated by 40 [mm] and the object is placed 75 [mm] before the first lens. I've to...

1,2,3. Homework Statement I tried to derive the length contraction using the Lorentz transformation matrix and considering 2 events. I reached the correct result but there's a step that I had to assume that I don't understand. Consider a ruler of length L along the x-axis for an observer at...

Suppose we pick a matrix M\in M_n(ℝ) s.t. all its eigenvalues are strictly bigger than 1. In the question here the user said it induces some norm (|||⋅|||) which "expands" vector in sense that exists constant c∈ℝ s.t. ∀x∈ℝ^n |||Ax||| ≥ |||x||| . I still cannot understand why it's correct. How...

Homework Statement Let ##f : \mathbb{R}^n \rightarrow \mathbb{R}^m## be a linear function. Suppose that with the standard bases for ##\mathbb{R}^n## and ##\mathbb{R}^m## the function ##f## is represented by the matrix ##A##. Let ##b_1, b_2, \ldots, b_n## be a new set of basis vectors for...

Homework Statement Show that the matrix ##A## is of full rank if and only if ##ad-bc \neq 0## where $$A = \begin{bmatrix} a & b \\ b & c \end{bmatrix}$$ Homework EquationsThe Attempt at a Solution Suppose that the matrix ##A## is of full rank. That is, rank ##2##. Then by the rank-nullity...

I saw this somewhere but I think it is wrong... I already read Griffiths' "Introduction to Particle Physics" (the 1st edition) from the page 216 to the page 222 (chapter of Quantum Electrodynamics - section "Solution to the Dirac Equation") and I didn't understood why was there the imaginary...

It's from the chapter on Matrix Inverses... imgur link: http://i.imgur.com/8OhFzgi.jpg This is the entirety of the exercise. It's not following on from or setting anything else up. That's just number 42...what can I do with this?

Hi, I found a statement without a proof. It seems simple enough, but I am having trouble proving it because I am not positive about induced matrix norms. The statement is that $$||A^k|| \leq||A||^{k}$$ for some matrix A and positive integer k. I have found that the norm of a matrix is the...

Homework Statement Homework Equations and attempt at solution I think I got the ground state, which can be expressed as |\Psi \rangle = \prod_{k}^{N}\hat{a}_{k}^{\dagger} |0 \rangle . Then for the density matrix I used: \langle...

How many degree of liberty exist, actually, in a matrix 2x2 ? I think that is three! Because the conic equation can be wrote like this: \begin{bmatrix} A & B\\ C & D \end{bmatrix} :\begin{bmatrix} x^2 & xy\\ yx & y^2 \end{bmatrix} + \begin{bmatrix} E\\ F \end{bmatrix} \cdot \begin{bmatrix}...

Homework Statement My question is regarding part (e), I just gave all the questions for reference. Homework EquationsThe Attempt at a Solution These are the coupled equations I should solve (from part d) My issue is using ode45 to get ##C_{A}(t)##, ##C_{P}(t)##, and ##T(t)##. Here is my...

Homework Statement Let A and B be n x m matrices, and λ and μ be real numbers. Prove that: (λA+μB)^T = λA^T+μB^t Homework Equations :/ The Attempt at a Solution I'm struggling to start here. If there was no λ and μ, I think I'd be able to reasonably solve this. How do I show that these...

Most definitions of a matrix that I have seen involve entries that are elements of a field. What if I have a unorderd set with no operations defined on it, say a set of different colored marbles or a set of historical events. Can I have a matrix whose entries are elements of such a set?

Homework Statement Find 2x2 matrices A and B, all of whose entries are \begin{align} &\geq 0 \end{align}, such that A^-1 and B^-1 exist, but (A+B)^-1 does not exist. Homework Equations The insverse is defined as 1/determinat(matrix) * adj(matrix) Otherwise shown as...

Hi, rapid fire posting in this subforum I know, sorry if that's annoying. Let me know if I should space my posts out a bit more. Here's an image of the solution to a worked example (from Intro to Linear Algebra 4th by Strang) here's the imgur link: http://i.imgur.com/IG6r15H.jpg I cannot...