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. C

    Partial trace of density matrix

    I am unsure how to (mathematically) do the partial trace of a density matrix so that I can find the expectation value of an observable. I am working on a model similar to the Jaynes cummings model. My density matrix is of the form; \rho = [\rho_{11}, \rho_{12}, \rho_{21}, \rho_{22}]...
  2. E

    Trace element analysis techniques

    I have a polymer sample and am interested in qualitatively determining the sulfur (sulfate group) content. Which analytical technique should I consider using? This polymer is not soluble in water but is soluble in THF and toluene. Thanks.
  3. X

    Trace and its square of mixed state density operator using integral

    Homework Statement I want to show that tr\left(\hat{\rho}_{mixed}\right)=1 tr\left(\hat{\rho}_{mixed}^{2}\right)<1 when \hat{\rho}_{mixed}=\frac{1}{2\pi}\int_{0}^{2\pi}d \alpha \hat{\rho}(\psi) Homework Equations tr\left(\psi\right)= \sum_{n}\langle n|\psi|n\rangle...
  4. Einj

    Problem with SU(3) generators's trace

    Hi everyone. I'm not sure this is the correct section for this topic and if not my apologiez. I'm studying SU(3) and my professor wrote down the following equality: $$Tr\left(\left[ T^a_8,T^b_8\right] T^c_8\right)=i\frac{3}{2}f^{abc}$$ where Ts are generators of the adjoint...
  5. I

    Trace as a product of operators

    I'm confused about index calculation in eq. 8.25, Mandl QFT textbook. Can anyone give me a detailed explanation showing the equality below? X=\frac{1}{2}A_{\delta \alpha}^+(\bf{p'})\Gamma_{\alpha \beta}(\bf{p'})A_{\beta \gamma}^+(\bf{p})\widetilde{\Gamma}_{\gamma\delta}...
  6. sunrah

    Expectation value via trace

    Homework Statement given \mid \psi \rangle = \frac{1}{\sqrt{2}} (\mid1\rangle + \mid2\rangle ) where \mid1\rangle, \mid2\rangle are orthonormal calculate i)density operator ii) \langle A \rangle where A is an observable Homework Equations The Attempt at a Solution i) \rho = \frac{1}{2}...
  7. C

    Partial trace of a density matrix?

    Hi, I'm working on a modified version of the Jayne's Cummings model and am a little confussed. I have: -Taken modified version of JCM Hamiltonian in Schrodinger picture. -Used Von Neumann equation to get evolution of density matrix -Converted to Wigner function. I want to run...
  8. D

    Relationship between Trace and Determinant of Unitary Matrices

    Homework Statement If U is a 2 x 2 unitary matrix with detU=1. Show that |TrU|≤2. Write down the explicit form ofU when TrU=±2 Homework Equations Not aware of any particular equations other than the definition of the determinant and trace. The Attempt at a Solution I have...
  9. V

    Proving the Trace of a Tensor Equation Using Dot and Cross Products

    Homework Statement Could someone guide me to proof following equation. For any A is a tensor; a, b, c are vectors. Proof that: Tr (A) a.(b x c) = Aa.(b x c) + a.(Ab x c) + a. (b x Ac) with (.) is dot product, and (x) is cross product of vector Homework Equations The...
  10. S

    Trace of product the of tensors

    Let A, B be matrices with components Aμν , Bμν such that μ, ν = 0, 1, 2, 3. Indices are lowered and raised with the metric gμν and its inverse gμν. Find the trace of ABA-1 in component form? Since A and B are generalized versions of tensors, finding their inverse becomes very tedious if we try...
  11. J

    Trace - Integration - Average - Tensor Calculus

    Hi Let D be an anisotropic tensor. This means especially, that D is traceless. \mathrm{tr}(D) = 0 Apply the representating matrix of D to a basis vector S , get a new vector and multiply this by dot product to your basis vector. Than you got a scalar function. Now integrate this...
  12. J

    One-point compactification of space of matrices with positive trace

    one-point compactification of space of matrices with non-negative trace Hi I'm a physicist and my question is a bit text-bookey but it is also part of the proof that the universe had a beginning...so could I ask anyway...You got q which is a continuous function of a 3 by 3 matrix where if any...
  13. O

    Find the Trace and Determinant

    T:V -> V is linear. V is finite vectorspace of dimension m^2. T(M) = AMB where M is an mXm matrix and A, B are two fixed mXm matrices. I want to find the trace and determinant of this transformation. In the case where B is the indentity, I can show that the trace is m*tr(A) and the...
  14. F

    Components of the trace operation

    I'm currently reading "Introduction to tensors and Group Theory for Physicists". I'm stuck on a question posed on dual spaces. The author gives the trace as an example of a linear functional on the vector space M_n(ℝ) (n x n matrices with real entries) and then asks how one would find the...
  15. Demon117

    Proving a matrix exponential determinant is a exponential trace

    Homework Statement Prove that for any matrix A, the following relation is true: det(e^{A})=e^{tr(A)} The Attempt at a Solution PROOF: Let A be in Jordan Canonical form, then A=PDP^{-1} where D is the diagonal matrix whose entries are the eigenvalues of A. Then...
  16. L

    Determinant as a function of trace

    for dimension 2, the following relation between determinant and trace of a square matrix A is true: det A=((Tr A)2-Tr (A2))/2 for dimension 3 a similar identity can be found in http://en.wikipedia.org/wiki/Determinant Does anyone know the generalization to dimension 4 ? lukluk
  17. M

    Partial Trace Q: Meaning & Info Explained

    Hi, I am not able to understand something about partial tracing. We have a quantum state \rho_{AB}. The Hilbert Space is H_{A}\otimes H_{B}. For some observable A in H_{A}, we have Tr_{A}(\rho_{A}A)=Tr_{AB}(\rho_{AB}(A\otimes 1)) =\sum\sum<a_{j}, b_{k}|\rho_{AB}(A\otimes 1_{B})|a_{j}, b_{k}>...
  18. jfy4

    Understanding the Trace of the SEM Tensor

    Hi, Let T_{\alpha\beta} be the stress-energy momentum tensor. What does g_{\alpha\beta}T^{\alpha\beta} mean? I have always thought of the Ricci tensor and the SEM as the same thing essentially, but the Ricci scalar essentially assigns a number to the curvature of the manifold, what does T...
  19. E

    Question on the trace of two matrices

    How does knowing that two matrices anticommute AB=-BA and that A^2=1 and B^2=1 help me to know how to find the trace of the matrices. I am supposed to show that their traces equal each other which equals 0 but I am not sure exactly how the given information helps me determine the trace?
  20. A

    A trace between a dog and a rabbit

    Homework Statement A dog is at a distance L due north of a rabbit. He starts to pursue the rabbit and its motion always points to the rabbit. Given that the rabbit keeps running due east with a constant speed v and the dog's speed is a constant u, where v<u. Find the time that the dog catches...
  21. maverick280857

    Trace of higher powers of Density Matrix

    Hi, The Quantum Liouville Equation is \dot{\rho} = \frac{i}{\hbar}[\rho, H] where the dot denotes the partial derivative with respect to time t. We take \hbar = 1 hereafter for convenience. Tr(\dot{\rho}) = 0 Consider Tr(\rho^2) Differentiating with respect to time...
  22. E

    Is There a Discrepancy in Matrix Trace Derivative Rules?

    Hope this is the right section. I'm having trouble ironing out an apparent inconsistency in matrix trace derivative rules. Two particular rules for matrix trace derivatives are \frac{\partial}{\partial\mathbf{X}} Tr(\mathbf{X}^2\mathbf{A})=(\mathbf{X} \mathbf{A}+\mathbf{A} \mathbf{X})^T...
  23. N

    Proving that A=0 When tr(A^2)=0

    i need to prove that if tr(A^2)=0 then A=0 we have a multiplication of 2 the same simmetrical matrices why there multiplication is this sum formula A*A=\sum_{k=1}^{n}a_{ik}a_{kj} i know that wjen we multiply two matrices then in our result matrix each aij...
  24. A

    Why Rank is the Trace of a Projection

    Why is the Trace of a projection is its Rank. Thank you
  25. L

    Trace of a Matrix: Definition & Analysis

    In many books and also in wikipedia, the Trace of a matrix is defined as sum of its diagonal elements. For a general matrix, it does not make much sense, as any element is as important any other element. An alternative definition (in wikipedia for example) is that the Trace of the matrix...
  26. T

    General Tensor contraction: Trace of Energy-Momentum Tensor (Einstein metric)

    Okay so I have: Eqn1) Tij=\rhouiuj-phij = \rhouiuj-p(gij-uiuj) Where Tij is the energy-momentum tensor, being approximated as a fluid with \rho as the energy density and p as the pressure in the medium. My problem: Eqn2) Trace(T) = Tii = gijTij = \rho-3p My attempt: Tr(T) = Tii...
  27. C

    Plutonium - the last ironic trace of civilization?

    At least 3 million years are required to form a geological strata (of say 7 ft?). What remanents of a culture might survive? Plastic pieces; or nothing at all? Our world has hundreds of tons of plutonium; plus all reactors forming plutonium. The halve life of plutonium is very long...
  28. D

    Trace Theorems and Dirac Matrices

    I think I'm missing something real simple on trace theorems and Dirac matrices, but am just not seeing it. In the Peskin and Schroeder QFT text on page 135 we have: gamma^(mu)*gamma^(nu)*gamma_(mu) = -2*gamma^(nu) But, why can't we anti-commute and obtain the following...
  29. V

    Derivative of metric tesor and its trace

    I would like to ask, how these identities are true \partial_{\mu}(-g)=(-g)g^{\alpha\beta}\partial_{\mu}g_{\alpha\beta} and \partial_{\mu}g^{\alpha\beta}=-g^{\alpha\lambda}g^{\beta\rho}\partial_{\mu}g_{\lambda\rho} Sorry I meant" derivative of metric tensor and its determinant", I was able to...
  30. B

    Trace of 3x3 Matrix | Linear Function | Basis Set Representation

    Homework Statement Let X denote the set of all real symmetric 3x3 matrices. The trace of a matrix, tr(x) is defined as the sum of the diagonal components and is a linear function. Define L(x) = tr(x), where x refers to X. Find the representation of this operator with respect to a basis set for...
  31. O

    Derivative (mimization) of matrix trace

    Given a function f(X)= Tr(X'AX) - 2Tr(X'BC), with X' denoting matrix transpose, I'm supposed to find the expression used to miminize the function with respect to X. The derivatives should be used, but I'm not sure how to proceed. Any help is appreciated.
  32. C

    Trace of this yang-mills operator

    Hi there, I'm trying to compute the trace of an operator found here: http://inspirebeta.net/record/360247 (eq 7.5) I'm not going to make you read the article, so i state the problem: I have the following operator in a Yang-.Mills theory, using the background field method...
  33. A

    How can I show that trace is Invariant under the change of basis?

    How can I show that trace is Invariant under the change of basis?
  34. E

    Prove that dirac matrices have a vanishing trace

    Not a Homework problem, but I think it belongs here. Homework Statement Consider four dirac matrices that obey M_i M_j + M_j M_i = 2 \delta_{ij} I knowing the property that Tr ABC = Tr CAB = Tr BCA show that the matrices are traceless. Homework Equations Tr MN = Tr NM The Attempt...
  35. R

    Trace of momentum-space propagator

    The integral: \int \Pi_k d\phi_k e^{-\phi_i A_{ij} \phi_j} is a Gaussian and is equal to: (\pi)^{n/2}\sqrt{det(A^{-1})}= (\pi)^{n/2} e^{\frac{1}{2}Tr ln A^{-1}} Now usually A is a diagonal matrix that represents the Lagrangian (so that the sum over i and j collapses to a sum just over i...
  36. A

    Determinant and trace of matrix ( HELP)

    FInd the determinant of the following matrix? 4,-4,-8 -2, 2, 6 0, 0,-1 Heres my attempt 4.(2x(-1)- 6x0) -(-4).((-2)x(-1) - 6x0) +(-8).((-2)x0-2x0) which goves: 4.(-2)+4(2) -8 = 0 is this correct?? Im also asked to find the trace? What is this and how do i find it...
  37. A

    Lattice Field Theory - QCD. Trace?

    Hi guys, So I'm working on this project to simulate QCD using a computer using (hybrid) monte carlo. I follow the majority of what I've read thus far, though there are a few things I'm uncertain about. Firstly, the Wilson loop is often written as U_{P} and the invariant gauge action is...
  38. S

    Dimension of subspace of trace of matrix

    Let V=Mn(k) be a vector space of matrices with entries in k. For a matrix M denote the trace of M by tr(M). What is the dimension of the subspace of {M\inV: tr(M)=0} I know that I am supposed to use the rank-nullity theorem. However I'm not sure exactly how to use it. I know that the trace is...
  39. J

    Nullity of the trace of a matrix

    Homework Statement What is the nullity of the trace (A), A is an element of all nxn square matrices. The Attempt at a Solution the null space would be when the sum of the diagonal is equal to 0. So the Σaii for i=1 to n must equal 0 which would be when aii = -aii. Therefore the...
  40. R

    Inductance of a Trace: Qualitative Analysis

    If I have a trace on a PCB that is wider on the left side and then tapers down to a narrower trace on the right side, why is it that the inductance is greater on the narrower side? I realize there are equations that describe this behavior but I'm just trying to get a qualitative understanding of...
  41. S

    Density Operators, Trace and Partial Trace

    I have some math questions about quantum theory that have been bugging me for a while, and I haven't found a suitable answer in my own resources. I'll start with the Trace operation. Question A) My understanding is that if we take system A and perform the partial trace over system B, we...
  42. S

    A stupid question on norm and trace of fields

    so i came up with a proof that..well.. Let L/K be a field extension and we have defined the norm and trace of an element in L, call it a, to be the determinant (resp. trace) of the linear transformation L -> L given by x->ax. Now it's well known that the determinant and trace are the...
  43. J

    What is the Trace of Density of States?

    regarding the density of states: how I GET THE FOLLOWING EQUALITY? \langle E_n\mid \delta(E-\widehat{H}) \mid E_n \rangle = \sum_n \delta(E-E_n)
  44. D

    Trace of the fundamental commutation relation

    Hi. So I have learned that this holds for the trace if A and B are two operators: \text{Tr}(AB)=\text{Tr}(BA). Now I take the trace of the commutator between x and p: \text{Tr}(xp)-\text{Tr}(px)=\text{Tr}(xp)-\text{Tr}(xp)=0. But the commutator of x and p is i\hbar. Certainly the trace of...
  45. A

    Polarization of EM wave - does the E vector trace an ellipse w.r.t space as well ?

    Let us consider the Electric field components of a polarized EM wave . [PLAIN]http://www.cdeep.iitb.ac.in/nptel/Electrical%20&%20Comm%20Engg/Transmission%20Lines%20and%20EM%20Waves/graphics/CHAP%204__255.png. Now if we fix the value of z (for convenience take z=0) and consider the locus of...
  46. K

    Relationship between determinant and trace

    Hi... We have all seen the equation det(M)=exp(tr(lnM)). I was taught the proof using diagonalisation. I was wondering if there was a proof for non-diagonalisable matrices also.
  47. B

    Proof of trace of density matrix in pure/mixed states

    Can someone help me prove that tr(\rho^2) \leq 1 ? Using that \rho = \sum_i p_i | \psi_i \rangle \langle \psi_i | \rho^2 = \sum_i p_i^2 | \psi_i \rangle \langle \psi_i | tr(\rho^2) = \sum_{i, j} p_i^2 \langle j | \psi_i \rangle \langle \psi_i | j \rangle Where do I go from here? Thanks guys.
  48. D

    Trace, determinant, and eigenvalues 3x3

    Use the trace and determinant to compute eigenvalues. I know how to do this with a 2x2 but not sure how to do it with a matrix of nxn where n>2. \begin{bmatrix} \frac{1}{2} & \frac{1}{3} & \frac{1}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}\\ \frac{1}{4} & \frac{1}{3} & \frac{2}{5}...
  49. Q

    Calculating trace with slashed item

    Homework Statement I was asked to find Tr q (p + m) q (p + m) Homework Equations Tr p q = 4pq The Attempt at a Solution If I expand it as Tr (p q p q + m q p q + m q q p + (m^2)(q)^2 ), although Tr Π(odd number of gamma matrices) = 0, since q p q and similar terms are not square...
  50. J

    Calculating Derivatives and Traces: A Guide to Solving for det(I + tA) = tr(A)

    Hey guys, any hints on how to show that \frac{d}{dt}|t=0 det(I + tA) = tr(A) ? I did it for 2x2 but I can't figure out a generalization. Thanks
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