What is Trace: Definition and 197 Discussions

Transition Region and Coronal Explorer (TRACE) was a NASA heliophysics and solar observatory designed to investigate the connections between fine-scale magnetic fields and the associated plasma structures on the Sun by providing high resolution images and observation of the solar photosphere, the transition region, and the corona. A main focus of the TRACE instrument is the fine structure of coronal loops low in the solar atmosphere. TRACE is the fourth spacecraft in the Small Explorer program, launched on April 2, 1998, and obtained its last science image on 21 June 2010 23:56 UT.The satellite was built by NASA's Goddard Space Flight Center. Its telescope was constructed by a consortium led by Lockheed Martin's Advanced Technology Center. The optics were designed and built to a state-of-the-art surface finish by the Smithsonian Astrophysical Observatory (SAO). The telescope has a 30 cm (12 in) aperture and 1024×1024 CCD detector giving an 8.5 arc minute field of view. The telescope is designed to take correlated images in a range of wavelengths from visible light through the Lyman alpha line to far ultraviolet. The different wavelength passbands correspond to plasma emission temperatures from 4,000 to 4,000,000 K. The optics use a special multilayer technique to focus the difficult-to-reflect EUV light; the technique was first used for solar imaging in the late 1980s and 1990s, notably by the MSSTA and NIXT sounding rocket payloads.

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  1. E

    Trace distance between two probability distributions - prove

    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}...
  2. E

    Trace Distance in quantum mechanics

    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...
  3. V

    How do I simplify the calculation of this trace involving six gamma matrices?

    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: $$...
  4. P

    Orthogonality of inner product of generators

    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.
  5. T

    Computing Cylinder Pressure from Temperature Trace

    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...
  6. N

    Trace of Eq.(A.4) - Can Anyone Help?

    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...
  7. billyp245

    Double Pendulum in C (Trace Motion Real-Time + gif output)?

    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...
  8. Y

    How to measure fluorescence intensity time trace?

    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...
  9. C

    [Semiclassical physics] 1D box trace formula

    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)...
  10. B

    Trace of a particular matrix product

    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}##...
  11. G

    How to trace over spinor indices?

    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)...
  12. R

    How to measure trace impedance

    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?
  13. W

    Finding the trace of the inverse of a 2x2 matrix in (mod 26)

    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...
  14. G

    Proof of Trace Orthogonality Relation for Matrices $\Gamma^A$

    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...
  15. R

    Understanding Traceless Proof for Gamma Matrices

    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) &=...
  16. J

    Trace of Symmetric Matrix Proof

    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...
  17. kq6up

    Is the trace of an outer product always equal to 1?

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

    Linear algebra: trace and dual space exercise

    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...
  19. J

    Can the trace be expressed in terms of the determinant?

    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?
  20. J

    How write a matrix in terms of determinant and trace?

    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(...
  21. J

    Trace Inequality Tr(ABAB)

    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!
  22. E

    What is the practical application of the trace of a square matrix?

    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...
  23. S

    Understanding the Relation Between Determinant and Trace in Physics Texts

    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...
  24. C

    Write trace of AB* as summation

    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...
  25. H

    Why we use the trace calculation?

    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.
  26. C

    Trace of the spin matrix of spin-1

    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...
  27. L

    Area under v vs t sinusoidal trace

    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...
  28. J

    Trace of elements in a finite complex matrix group is bounded

    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...
  29. ajayguhan

    Complex matrix, trace.

    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...
  30. D

    Trace of operator with continuous spectrum

    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...
  31. S

    Density function for the trace of a singular value matrix

    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...
  32. T

    What is the Connection Between Matrix Trace and Endomorphism?

    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...
  33. fluidistic

    Trace of the stress-energy tensor

    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...
  34. fluidistic

    A few very basics questions regarding 4 and 3 momenta and the trace of

    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...
  35. U

    Trace of Position and Momentum Operator

    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] =...
  36. U

    Trace of Matrix Product as Scalar Product

    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...
  37. V

    Calculating an expression for trace of generators of two Lie algebra

    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...
  38. sunrah

    Relationship between trace and phase space

    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...
  39. marellasunny

    Why is the trace of jacobian=the divergence?

    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...
  40. P

    Dirac trace in D dimension with gamma_5

    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?
  41. B

    Trace of a product of gamma matrices

    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...
  42. S

    Linear algerba: trace of square matrix is a linear functional

    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...
  43. G

    Matrix trace minimization and zeros

    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
  44. H

    Finding a row vector and calculating the trace of a matrix

    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...
  45. twoski

    MARIE Program Trace for Fetch-Execute Cycle with Register Transfer Notation

    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...
  46. N

    Trace reverse perturbation

    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##
  47. S

    How unitary change of basis related to Trace?

    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...
  48. C

    Partial trace of the density matrix

    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...
  49. N

    Find the Kernel of the Trace of a Matrix

    Homework Statement Let F : Mnn(R) → R where F(A) =tr(A). Show that F is a linear transformation. Find the kernel of F as well as its dimension. What is the image of F? Homework Equations The Attempt at a Solution I have shown that it is a linear transformation. But I am not...
  50. I

    Mathematical Quantum Statistics: Why is A*rho of trace class?

    Hi, as we know a density operator \rho is defined to be a non-negative definite operator of trace class (with trace 1). We also know that for a given observable A, which is a (possibly unbounded) self-adjoint operator, the expectation value can be calculated as \operatorname{tr}(A \cdot...
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