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. binbagsss

    I Exploring the Trace of a Fourth Rank Tensor in Index Notation

    What is the general expression for the trace of a fourth rank tensor? Do you sum over possibilities of contractions with some factor? So, for instance, for the Riemann tensor, it is given by: $\eta_{ab}\eta_{cd}R^{acbd}$ due to these being independent contractions due to the symmetry...
  2. Euge

    POTW Comparing Rank and Trace of a Matrix

    Let ##M## be a nonzero complex ##n\times n##-matrix. Prove $$\operatorname{rank}M \ge |\operatorname{trace} M|^2/\operatorname{trace}(M^\dagger M)$$ What is a necessary and sufficient condition for equality?
  3. Euge

    POTW A Linear Operator with Trace Condition

    Let ##V## be a finite dimensional vector space over a field ##F##. If ##L## is a linear operator on ##V## such that the trace of ##L\circ T## is zero for all linear operators ##T## on ##V##, show that ##L = 0##.
  4. yucheng

    I Partial trace and the reduced density matrix

    From Rand Lectures on Light, we have, in the interaction picture, the equation of motion of the reduced density matrix: $$i \hbar \rho \dot_A (t) = Tr_B[V(t), \rho_{AB}(t)] = \Sigma_b \langle \phi_b | V \rho_{AB} -\rho_{AB} V | \phi_b \rangle = \Sigma_b \phi_b | \langle V \rho_{AB} | \phi_b...
  5. Melbourne Guy

    Good gas to trace lasers in a vacuum

    I'm excitedly close to the end of proofing my latest novel, and as happens, tweaking of passages occurs. In this instance, a fleet of warships travel together in a collective warp space bubble that has inconsistent gravity, especially around the edges. To facilitate shuttle transfers between...
  6. S

    Trace on the screen of a C.R.O. when a deflection coil is rotated

    My answer is when the small coil is rotated, there will be change in magnetic flux in it. Initially (before rotated), the trace in c.r.o will be sinusoidal. After it is rotated, the trace will still be a sinusoidal (but I am not sure whether the amplitude will be the same). But the answer key...
  7. E

    A Trace of Numerator in QED vacuum polarization

    Sorry I just typed out my query .For some reason I can't seem to find the buttons for attaching files on this thread. When writing the QED vacuum polarization loop, the numerator ,consisting momenta slashed + m from the fermion propagators and the two gamma matrices, has a trace over all of it...
  8. K

    Could our body deal with trace amounts of unknown chemicals?

    Even if how careful we are, I think it is hard for us to avoid TRACE amount of unknown chemicals that get into our body unconsciously, through different pathway, our mouth, our nose and our eyes or our wound. Could our body handle these foreign chemicals in trace amount and keep us healthy? Does...
  9. H

    A Why are we allowed to use the trace cyclicity here?

    Hi Pf i am reading this article: pillet.univ-tln.fr/data/pdf/KMS-states.pdf I know that the trace cyclicity can be used when there is a product of matrices. But here we have operators (an hamiltonian , an operator which can be the position operators) . the author take the trace of a product. is...
  10. O

    I Need some help with trace calculation in an index theorem

    I hate to create a thread for a step in a calculation, by I don't know what else to do. I'm having a lot of trouble reproducing E. Weinberg's index calculation (found here https://inspirehep.net/literature/7539) that gives the dimension of the moduli space generated by BPS solutions in the...
  11. N

    A Trace of the inverse of matrix products

    Hello, I am puzzled about the following condition. Assume a matrix A with complex-valued zero-mean Gaussian entries and a matrix B with complex-valued zero-mean Gaussian entries too (which are mutually independent of the entries of matrix A). Then, how can we prove that...
  12. PainterGuy

    Proving the results for the trace of a matrix

    Hi, I was trying to do the following problem. I was able to do the part in pink highlight (please check "My attempt") but the part in orange highlight makes no sense to me. I'd really appreciate if you could help me to solve the part in orange. Thank you! My attempt: The solution presented...
  13. Dukon

    B Trace methane in Uranus and Neptune, condensing into diamonds

    Uranus and Neptune have roughly an 80-19-1 $H_2$-$He_4$-$CH_4$ mixture. (Of course actual percentages vary but those are rough values for sake of this simplification argument -- unless their actual values are critical to the answer.) Uranus temperature is reported as 2K while Neptune is 20K...
  14. LCSphysicist

    I What is the Minkowski metric tensor's trace?

    I am trying to follow the rule, that is, raising an index and the contract it. Be ##g_{\mu v}## the metric tensor in Minkowski space. Raising ##n^{v \mu}g_{\mu v}## and then, we need now to contract it. Now, in this step i smell a rat (i learned this pun today, hope this mean what i think this...
  15. J

    Why are PCB Trace and Layer Spacings Based on Voltage?

    I understand that PCB trace spacing is typically based on a minimum found in certain standards and that is voltage based. If the breakdown strength of the dielectric is based on e-field intensity, wouldn't it be beneficial to actually consider the material properties and make it based on a...
  16. Haorong Wu

    How to calculate a trace of a tensor?

    The textbook gives some examples for ##P_1 \left ( cos \theta \right)##, ##P_2 \left ( cos \theta \right)##, and ##P_3 \left ( cos \theta \right)##. The procedure is clear. The key to the problem is to find the symmetric traceless tensor for the proper rank. I first construct a symmetric...
  17. JD_PM

    Solving Computing a Trace Properties: Any Help Appreciated

    Properties 1) If ##(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})## contains an odd number of ##\gamma##-matrices $$Tr(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})=0$$ 2) $$Tr(\not{\!A}\not{\!B})=4AB$$ 3) $$Tr(\not{\!A}\not{\!B}\not{\!C}\not{\!D})=4\Big(...
  18. JD_PM

    I Why does the trace show up while computing unpolarized cross sections?

    I've been studying different scattering processes (from Mandl & Shaw QFT's book, chapter 8) and there's always a common step I do not understand: the showing-up of the trace. Let me give two specific examples. Lepton Pair Production ##( e^+ (\vec p_1, r_1) + e^- (\vec p_2, r_2) \rightarrow l^+...
  19. D

    I Trace of the stress energy tensor

    The stress energy tensor has many forms based on the type of matter you are describing, dust, fluid, perfect fluid... is it true that the trace of all of these matter situations is invariant?
  20. beefbrisket

    I Taking a partial trace of a multipartite state for measurement

    I am attempting to understand how POVMs fit in with quantum measurement, and I think I am getting tripped up in notation when it comes to multipartite systems. The situation is as follows: System: \rho_A Measurement instrument: \rho_B = |\phi\rangle\langle\phi| (pure state) The multipartite...
  21. RicardoMP

    A Trace of a product of Dirac Matrices in a Fermion loop

    I'm working out the quark loop diagram and I've drawn it as follows: where the greek letters are the Lorentz and Dirac indices for the gluon and quark respectively and the other letters are color indices. For this diagram I've written...
  22. G

    Fluke PM3384A combiscope trace fault

    I’ve just bought this useful looking 100 MHz scope in the hope of repairing it. When I switch to digital mode, the trace disappears off-screen on all four channels. 1 and 3 are up high, 2 and 4 down low. Sometimes the top or bottom of the trace is visible, and it flickers up and down the...
  23. Haorong Wu

    Trace of a subsystem of a two qubit system

    Consider the first qubit (subsystem A): First, the density operator for the system AB is ## \rho ^{AB} =\frac { \left |00 \right > \left <00 \right |+ \left |01 \right > \left < 01\right |+\left | 11\right > \left < 11\right | } 3 ##. Then, the reduced density operator of subsystem A is ##...
  24. M

    MHB Is the Kernel of Trace Equal to the $F$-Subspace of $V_\sigma$?

    Hey! :o Let $K/F$ be a finite Galois extension and let $G= \operatorname{Gal}(K/F)$. For each $\sigma\in G$ we define $V_{\sigma}=\{\sigma (b)-b:b\in K\}$. Show that $V_{\sigma}$ is $F$-subspace of $\ker \operatorname{Tr}_{K/F}$. Show that $K/\mathcal{F} (\langle \sigma \rangle) \cong...
  25. A

    I Yang-Mills Lagrangian: Is ##F^{\mu \nu}F_{\mu \nu}## a Number?

    I am sorry for asking this stupid question, but in the Yang-Mills lagrangian, there is a term ##Tr(F^{\mu \nu}F_{\mu \nu})##. Isn't ##F^{\mu \nu}F_{\mu \nu}## a number?
  26. BookWei

    I Casimir's trick / Evaluating trace

    Hi all, I am working on a project at the moment, and I have to evaluate the trace by using the Casimir's trick. The trace form is $$Tr[(\displaystyle{\not} P +M_{0})\gamma^{\mu}(\displaystyle{\not} P^{'} +M^{'}_{0})(\displaystyle{\not} p^{'}_{1} +m^{'}_{1})\gamma^{\nu}(\displaystyle{\not} p_{1}...
  27. L

    A Infinite matrices and the Trace function

    For finite matrices ##A## and ##B## we have Tr(AB)=Tr(BA) What happens in case of infinite matrices?
  28. S

    Pressure trace of a tank fed by a compressor

    G'Day All, This is my first post so please let me know if I have completed this form incorrectly, or missed a point of etiquette etc... 1. Homework Statement The problem is to determine the pressurisation rate of a tank being filled by a pipe connected to a compressor. Assumptions: Pipe...
  29. I

    A Second derivative of a complex matrix

    Hi all I am trying to reproduce some results from a paper, but I'm not sure how to proceed. I have the following: ##\phi## is a complex matrix and can be decomposed into real and imaginary parts: $$\phi=\frac{\phi_R +i\phi_I}{\sqrt{2}}$$ so that $$\phi^\dagger\phi=\frac{\phi_R^2 +\phi_I^2}{2}$$...
  30. VSayantan

    Trace of the Exponential of a Square Matrix

    Homework Statement Find the trace of a ##4\times 4## matrix ##\mathbb U=exp(\mathbb A)##, where $$\mathbb A = \begin {pmatrix} 0&0&0&{\frac {\pi}{4}}\\ 0&0&{\frac {\pi}{4}}&0\\ 0&{\frac {\pi}{4}}&0&0\\ {\frac {\pi}{4}}&0&0&0 \end {pmatrix}$$ Homework Equations $$e^{(\mathbb A)}=\mathbb P...
  31. A

    QM: Purity, two-outcome measurements

    Homework Statement Homework Equations in addition to those provided in the questions, I used the following: Tr(B) = sigma<x_j|B|x_j> purity = Tr(rho^2) The Attempt at a Solution [/B] I find calculating trace and purity very confusing. Am I on the right track with question 1? With...
  32. Tspirit

    Quantum Textbook for density matrix and trace?

    I want a QM textbook which introduces detail knowledge of density matrix and trace (i.e. the average), who can recommend one for me? Thank you.
  33. D

    Prove trace of matrix: Tr(AB) = Tr(BA)

    Homework Statement [/B] The trace of a matrix is defined to be the sum of its diaganol matrix elements. 1. Show that Tr(ΩΛ) = Tr(ΩΛ) 2. Show that Tr(ΩΛθ) = Tr(θΩΛ) = Tr(ΛθΩ) (the permutations are cyclic) my note: the cross here U[+][/+]is supposed to signify the adjoint of the unitary matrix U...
  34. T

    Proof of trace theorems for gamma matrices

    Hi, I'm currently going through Griffith's Particle Physics gamma matrices proofs. There's one that puzzles me, it's very simple but I'm obviously missing something (I'm fairly new to tensor algebra). 1. Homework Statement Prove that ##\text{Tr}(\gamma^\mu \gamma^\nu) = 4g^{\mu\nu}##...
  35. T

    I Is the trace of a matrix independent of basis?

    Hello, Just wondering if the trace of a matrix is independent of basis, seeing as the trace of a matrix is equal to the sun of the eigenvalues of the operator that the matrix is a representation of. Thank you
  36. P

    A Understanding the Use of Trace in QFT Lagrangian for Chiral Symmetry in QCD

    I have a question about the use of trace in QFT in general - more specifically the use of trace in the lagrangian in the effective theory concerning chiral symmetry in QCD. I am slowly trying to get a hang of everything, and most things i am able to calculate, but i still have som very specific...
  37. MrsM

    Linear Algebra: characteristic polynomials and trace

    The question is : Is it true that two matrices with the same characteristic polynomials have the same trace? I know that similar matrices have the same trace because they share the same eigenvalues, and I know that if two matrices have the same eigenvalues, they have the same trace. But I am...
  38. C

    B States as positive operators of unit trace

    I read that states are positive operators of unit trace - not elements of a vector space. Is it referring to quantum states or all classical states? I know operators are like minus, plus, square root and vectors are like rays in Hilbert space.. but why can't quantum states be vectors when in...
  39. O

    Sample preparation of powders for trace metal analysis

    When preparing a fine powder for trace metal analysis, is ashing prior to digesting unnecessary? If not, or not necessarily, where could I learn more about important considerations to determine whether of not any preparation prior to digestion is required? The one instance I have come across...
  40. J

    I What's the geometric interpretation of the trace of a matrix

    Hello, I was just wondering if there is a geometric interpretation of the trace in the same way that the determinant is the volume of the vectors that make up a parallelepiped. Thanks!
  41. munirah

    How to get the matrices in partial trace

    Homework Statement Good day, I want to ask the matrix that obtained from below formula and example. $$tr_A(L_{AB})=\sum_i [(\langle i|\otimes id)L_{AB}(|i\rangle\otimes id)]$$ this formula above can be represented as in matrix form below, $$tr_A(L_{AB})=...
  42. Arnd Obert

    I Density of states with delta function

    Hello, I'm stuck with this exercise, so I hope anyone can help me. It is to prove, that the density of states of an unknown, quantum mechanical Hamiltonian ##\mathcal{H}##, which is defined by $$\Omega(E)=\mathrm{Tr}\left[\delta(E1\!\!1-\boldsymbol{H})\right]$$ is also representable as...
  43. D

    A Expectation values and trace over the environment

    I've worked through a Stern Gerlach experiment for the Sx and Sz directions using the density matrix formalism to account for the environment. This shows a result which I think is correct but relies on decoherence to give the "actual" value. I'm not confident about the result though. Would...
  44. M

    Dirac Trace Algebra: Which Gamma Matrices Matter?

    Homework Statement This isn't a homework problem; it's just something I'm working on and I'm a little confused as to how to go about dealing with what I have. I have several traces of Dirac's gamma matrices, and I know that the trace of an odd number of gamma matrices is zero. So my first...
  45. S

    Trace of cathode ray in changing magnetic field.

    I know that cathode rays follow a helical pattern if they enter a uniform magnetic field at an angle less than 90 degrees. This leaves me with two questions. 1. If the cathode ray described above, hit a phosphor coated screen, would it show only 1 spot or a circle? 2. If the magnetic field was...
  46. D

    What is the trace of a second rank covariant tensor?

    What is the trace of a second rank tensor covariant in both indices? For a tensor covariant in one index and contravariant in another ##T^i_j##, the trace is ##T^k_k## but what is the trace for ##T_{ij}## because ##T_{kk}## is not even a tensor?
  47. G

    Zettili QM Problem on Trace of an Operator

    Homework Statement In Zettili's QM textbook, we are asked to find the trace of an operator |\psi><\chi| . Where the kets |\psi> and |\chi> are equal to some (irrelevant, for the purposes of this question) linear combinations of two orthonormal basis kets. Homework Equations...
  48. Matt atkinson

    Entanglement, projection operator and partial trace

    Homework Statement Consider the following experiment: Alice and Bob each blindly draw a marble from a vase that contains one black and one white marble. Let’s call the state of the write marble |0〉 and the state of the black marble |1〉. Consider what the state of Bob’s marble is when Alice...
  49. EsmeeDijk

    Setting the trace of a tensor equal to zero

    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...
  50. Safinaz

    How do I calculate the trace of SU(3) generators in the adjoint representation?

    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} ## ?