Trace Definition and 199 Threads

Homework Statement A tensor t has the following components in a given orthonormal basis of R3 tij(x) = a(x2)xixj + b(x2) \deltaij x2 + c(x2) \epsilonijk xk (1) where the indices i,j,k = 1, 2, 3. We use the Einstein summation convention. We will only consider orthogonal transformations...

Hi all, The trace of two SU(3) generators can be calculated by: ## T_{ij} T_{ji} = \frac{1}{2} ##, now how to calculate the trace of SU(3) generators: ## T_{il} T_{lk} T_{kj} T_{ji} ## ?

Homework Statement Let ##\{p_x\}## and ##\{q_x\}## be two probability distributions over the same index set ##\{x\}={1,2,...,N}##. Then, the trace distance between them is given by ##D(p_x,q_x):=\frac{1}{2} \sum_x |p_x-q_x|##. Prove that ##D(p_x,q,_x)=max_S |p(S)-q(S)|=max_S | \sum_{x \in S}...

Hi, I am trying to familiarize myself with the quantum mechanical trace distance and hit a brick wall. Thus, I would appreciate your help with the matter! I am reading up on trace distance using Nielsen, Chunang - Quantum Computation and Quantum Information and Bengtsson, Zyczkowski - Geometry...

Trace of six gamma matrices I need to calculate this expression: $$Tr(\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma^{\alpha}\gamma^{\beta}\gamma^{5}) $$ I know that I can express this as: $$...

Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.

I'm just trying to put together a very basic engine model with extremely limited combustion. Basically, I am modeling isentropic compression, combustion (using LHV to calculate the delta T), and isentropic expansion. At the moment, I am calculating a peak cylinder temperature under motoring of...

Is there anyone can give me a hand? http://arxiv.org/abs/astro-ph/0210603 When I read this paper I can not get Eq.(A.5) from (A.4). Why it is ##4\alpha##? If we take the trace of Eq.(A.4), why not it give us ##6\alpha##? Eq. (A.3): [Edited by a mentor to fix a small problem in the Latex...

I have written a C program to trace out the motion of a double pendulum, but am having difficulties in getting gnuplot (controlled from my c program) to trace out the paths of the masses (example video below). Thus far I have created the program such that it produces a number of png images at...

I understand that fluorescence intensity time trace is constantly monitor the fluorescence intensity and plot it over time. But the question is at which excitation wavelength? Also, what is the emitted wavelength that is being measured? I suppose it should be two particular wavelengths, but how...

Hey there. While studying the Single-particle Level Density, I encountered the example in the image below, referring to the One-dimensional Box problem. However, I do not understand what is it that he call's F(E), neither how does one go from that, to the density of states in Equation (3.64)...

Homework Statement Claim: If ##A \in \mathcal{M}_n (\mathbb{C})## is arbitrary, and ##D## is a matrix with ##\beta## in its ##(i-j)##-th entry, and ##\overline{\beta}## in its ##(j-i)##-th, where ##i \ne j##, and with zeros elsewhere, then ##Tr(AD) = a_{ij} \beta + a_{ji} \overline{\beta}##...

I would like to take the trace over spinorial indices of the following expression: (\gamma_{\mu}\gamma^{0})_{\alpha}^{\beta}=(\gamma_{\mu})_{\alpha}^{\gamma}(\gamma^{0})_{\gamma}^{\beta}. How do I go about doing this? I reckon I could expand the trace out (let's say I want to do this in 4D)...

Hi, I am working on a PCB that requires controlled trace impedance. I figured out width of the traces and layer stack thickness. How would I verify trace impedance once I get my PCB boards? What equipment do I need?

Homework Statement ...(2 , 3) A = ...(1 , 3) Find the trace of A^(-1) Homework Equations (a , b)......(d , -b) ...^(-1) = (ad-bc)^(-1)* (c , d).......(-c, a) The Attempt at a Solution .....(3 , -3) A^-1 = 9* .....(-1. 2) (In mod 26) ... (1, -1) = ... (-9...

I know that the matrices {\Gamma^{A}} obey the trace orthogonality relation Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B} In order to show that a matrix M can be expanded in the basis \Gamma^{A} in the following way M=\sum_{A}m_{A}\Gamma^{A} m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A}) is it enough to...

I'm reading through some lecture notes and there is a proof that the gamma matrices are traceless that I've never seen before (I've seen the "identity 0" on wikipedia proof) and I can't work out some of the steps: \begin{align*} 2\eta_{\mu\nu}Tr(\gamma_\lambda) &=...

Homework Statement Prove ##tr(AA^T)=tr(A^TA)=s## where ##s## is the sum of the squares of the entries of A I need help cleaning this up and I don't think my sigma notation is completely correct. The Attempt at a Solution I found the identity $$(AB)^T=B^TA^T$$then applying it to ##AA^T...

Is the trace of an outer product of a normalized state eq. (psi) equal to 1? Thanks, Chris Maness

Homework Statement . Let ##A \in \mathbb C^{m\times n}##. Prove that tr##(A^*A)=0## if and only if ##A^*A=0## (here ##0## obviously means the zero matrix). The attempt at a solution. By definition of the trace of a matrix, the implication ← is obvious. I am having problems proving...

Browsing in the wiki, I found those formulas: http://en.wikipedia.org/wiki/Determinant#Relation_to_eigenvalues_and_trace So, my doubt is: if is possible to express the determinant in terms of the trace, thus is possible to express the trace in terms of the determinant too?

Given the following: $$\\ \begin{bmatrix} A & 0\\ 0 & B\\ \end{bmatrix}$$ the eigenvalues is exactaly A and B. So analogously, is possible to write a matrix with only two elements, T and D, such that the trace is T and the determinant is D? I tried something: $$\\ \text{tr} \left(...

Here's the claim: Assume that A and B are both symmetric matrices of the same size. Also assume that at least other one of them does not have negative eigenvalues. Then \textrm{Tr}(ABAB)\geq 0 I don't know how to prove this!

I'm interested in the use/application of the trace of a square matrix? I am trying to get an intuitive feel for what it 'means' . . . Along the lines of: for a 2x2 matrix, the determinant represents the area of the parallelogram. I know it is the sum of the entries of the diagonal of a...

Hi. I am reading a physics text, and in one of the lines it uses the following relation: \mathrm{det}(\delta^\mu_\lambda +\frac{\partial \delta x^\mu}{\partial x^\lambda}) = 1 + \mathrm{Tr}\frac{\partial \delta x^\mu}{\partial x^\lambda} where \mu and \lambda are matrix elements, and...

i'm kinda confused regarding summation so I'm hoping someone can help me figure this out and explain to me why it is the way it is trace(AB*) = ? in summation form * = adjoint = conjugate and transpose = transpose and conjugate assume both matrices are square mx of same size n x n...

Dear all When we calculating the cross section of particles, we apply the trace calculation. This is not clear for me, why we use the trace calculation? Thank you all.

Spin-1 matrix Sx, Sy, Sz are traceless 3*3 matrix, and have the property ##[S_i, S_j] = i\epsilon_{ijk}S_k##, and we know that ##Tr(S_i^2) = 1^2+0^2+(-1)^2=2##. All of the above are independent of representation, of course, the trace of a matrix is representation-independent. so, if we want...

if I have a sinusoidal trace on an oscilloscope (v vs t) and I wanted to find the area under the wave form squared graph I could integrate the sqaured waveform with respect to t. but since i don't have the integration facility... is it fair to say that the area under the graph is proprtional...

Homework Statement Let G be a finite complex matrix group: G \subset M_{n\times n}. Show that, for g \in G, |\text{tr}(g)| \le n and |\text{tr}(g)| = n only for g = e^{i\theta}I. 2. The attempt at a solution Since G is finite, then every element g \in G has a finite order: g^r = I for some...

Homework Statement Let P be 2x2 complex matrice such that trace(p)=1 det(p)=-6 then trace(p^4-p^3) equals what...? Homework Equations Is there any formula for trace(A^n) The Attempt at a Solution Let the two eigen values be a,b a+b=1 a*b= -6 solving we get a=(1+i√23)/2...

Greetings, I must be missing something obvious but how is Tr{} defined exactly in case of contunuous spectrum operators? Everywhere I look I see it defined as a sum of [possibly infinite sequence of] eigenvalues. Is the following correct: Given Q = \int f(q) \left| q\right\rangle...

Hi All, It's been years since I have re-visited PF. I have an interesting problem today. I arises in a physical hypothesis testing problem: Problem Statement: what's the density function for the sum of singular values (trace of the singular value matrix) for a square, Gaussian matrix? My...

Consider the 4x4 matrices A = (1 2 3 4) (5 6 7 8) (9 10 11 12) (13 14 15 16)B= (1 2 3 4) (8 5 6 7) (11 12 9 10) (14 15 16 13) The question I was asked was the following: Show that there does not exist an endomorphism f: ℝ4 -> ℝ4 and basis 'a' and 'b' of R^4, such that A = a[f]a and B=b[f]b...

0. Homework Statement Hi guys, I must show that the trace of the stress energy tensor is zero. The definition of it is ##T^{\mu \nu }=\frac{1}{4\pi} \left ( F^{\mu \sigma } F^{\nu \rho} \eta _{\sigma \rho}-\frac{1}{4} \eta ^{\mu \nu } F^{\sigma \rho} F_{\sigma \rho} \right )##. 1. The...

Hi guys! I've got 2 extremely simple questions, hence a single thread. First, I want to know whether the conservation of the 4-momentum in a closed system implies the conservation of the energy and of the 3-momentum. Let's assume we consider 2 different times, ##t_i## and ##t_f##. Then...

Homework Statement Hi guys, I've not started a course on QM yet, but we are currently learning the maths used in QM. Show, by taking the trace of both sides show that finite dimensional matrix representations of the momentum operator p and the position operator x which satisfy [p, x] =...

Homework Statement Let V be the real vector space of all real symmetric n × n matrices and define the scalar product of two matrices A, B by (Tr (A) denotes the trace of A) Show that this indeed fulfils the requirements on a scalar product. Homework Equations Conditions for a scalar...

Suppose we have $$[Q^a,Q^b]=if^c_{ab}Q^c$$ where Q's are generators of a Lie algebra associated a SU(N) group. So Q's are traceless. Also we have $$[P^a,P^b]=0$$ where P's are generators of a Lie algebra associated to an Abelian group. We have the following relation between these generators...

So I noticed we can define entropy in two very different ways: 1) quantum mechanically S = -k Tr(\hat{\rho}\ln{(\hat{\rho})}) 2) classically S = -k \int \rho \ln{(\rho)} d\Gamma where Tr is the trace and d\Gamma = \frac{1}{h^{3N}N!}\prod_{i}^{N} d^{3N}q_{i}d^{3N}p_{i} is the phase...

What is the theoretical connection intuitively justifying that the trace of the jacobian=the divergence of a vector field? I know that this also equals the volume flow rate/original volume in the vector field but leaving that aside, what is the mathematical background behind establishing this...

I know the trace tr[\gamma_5 a\!\!\!/b\!\!\!/c\!\!\!/d\!\!\!/] in 4-dimensional space-time, how is the result of it in D dimension? Is it the same as in 4 dimension?

Homework Statement A proof of equality between two traces of products of gamma matrices. Tr(\gamma^\mu (1_4-\gamma^5) A (1_4-\gamma^5) \gamma^\nu) = 2Tr(\gamma^\mu A (1_4-\gamma^5) \gamma^\nu) Where no special property of A is given, so we must assume it is just a random 4x4 matrix. 1_4...

Lets define trace for each square matrix A its trace as sum of its diagonal elements, so tr_{n}(A)=\sum_{j=1}^{n}a_{j,j}. Now proove that trace is a linear functional for all square matrix. I would be happy to know what has to be true for anything to be a linear functional? If I...

Hello, :) I would like to minimize and find the zeros of the function F(S,P)=trace(S-SP’(A+ PSP’)^-1PS) in respect to S and P. S is symmetric square matrix. P is a rectangular matrix Could you help me? Thank you very much All the best GoodSpirit

Hi there, I've been having trouble with 2 algebra questions, I was hoping someone here could give me a hand. Homework Statement (i) Consider the function R2 → R2 defined by f (x, y) = (3x − 4y, x − 2y). Let v = (a, b) be the vector such that f (v) = (4, 6). Find the vector v and hence...

Homework Statement This question is being done using MARIE. Suppose initially AC = 1000, M[200] = FFFF, PC = 100. Trace the fetch execute (instruction) cycle of the following sequence of MARIE instructions using Register transfer notation (similar to Figure 4.14 of your text). Load 200...

From the tensor, ##\bar{h}^{ij}=h^{ij}-1/2\eta^{ij}h## Where, h=##h^i_i##, Prove that ##\bar{h}=-h##, Where, ##\bar{h}=\bar{h}^i_i##

Homework Statement Shanker 1.7.1 3.)Show that the trace of an operator is unaffected by a unitary change of basis (Equivalently, show TrΩ=TrU^{\dagger}ΩUHomework Equations I can show that via Shanker's hint, but I however can't see how a unitary change of basis links to TrΩ=TrU^{\dagger}ΩU...

Hi, I am trying to work out the atomic inversion of the Jaynes cummings model using the density matrix. At the moment i have a 2x2 matrix having used the Von neumann equation (technically in Wigner function in x and y). Each of my matrix elements are 1st order pde's describing the...