Symmetric Definition and 566 Threads

Symmetry (from Greek συμμετρία symmetria "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definition, and is usually used to refer to an object that is invariant under some transformations; including translation, reflection, rotation or scaling. Although these two meanings of "symmetry" can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art and music.
The opposite of symmetry is asymmetry, which refers to the absence or a violation of symmetry.

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  1. T

    How Does Angular Dependence Arise in a Spherical Symmetric Potential?

    For a spherical symmetric potential, the wavefunction can be expanded in terms of partial waves which is dependent on r and \theta . How would this be possible, when the potential only depends on distance from source? Classically, there's no quantity, that could have depend even on \theta .
  2. M

    Quadratic forms of symmetric matrices

    hi i just wanted a quick explanation of what a symmetric matrix is and what they mean by the quadratic form by the standard basis? (1) for example why is this a symmetric matrix [1 3] [3 2] and what is the quadratic form of the matrix by the standard basis? (2) also how would i go...
  3. K

    Bilinear forms & Symmetric bilinear forms

    1) Let f: V x V -> F be a symmetric bilinear form on V, where F is a field. Suppose B={v1,...,vn} is an orthogonal basis for V This implies f(vi,vj)=0 for all i not=j =>A=diag{a1,...,an} and we say that f is diagonalized. ============ Now I don't understand the red part, i.e. how does...
  4. Q

    Symmetric & Nondegenerate Tensor: Showing g is Invertible

    Homework Statement Let {e1, e2, e3} be a basis for vector space V. Show that the rank 2 tensor g defined by g=2E1*E2 + 2E2*E1+E1*E3+E3*E1 (where Ei are dual vectors and * is the tensor product) is symmetric and nondegenerate. Caculate g inverse. Homework Equations Um. lots of tensor stuff...
  5. E

    Is the Symmetric Tensor or Vector Equal to Zero Given a Specific Condition?

    Homework Statement If t_{ab} are the components of a symmetric tensor and v_a are the components of a vector, show that if: v_{(a}t_{bc)} = 0 then either the symmetric tensor or the vector = 0. Let me know if you are not familiar with the totally symmetric notation. Homework...
  6. H

    Electrostatic Self-energy of an arbitrary spherically symmetric charge density

    Homework Statement Find an expression for the electrostatic self-energy of an arbitrary spherically symmetric charge density distribution p(r). You may not assume that p(r) represents any point charge, or that it is constant, or that it is piecewise constant, or that it does or does not cut...
  7. Q

    Dimension of symmetric and skew symmetric bilinear forms

    Given the vector space consisting of all bilinear forms of a vector space V (let's call it B) it's very easy to prove that B is the direct sum of two subspaces, the subspace of symmetric and the subspace of skew symmetric bilinear forms. How would one go about determining the dimension of these...
  8. E

    Infinite Well Solutions: How Can Different Techniques Yield Contrasting Results?

    Homework Statement The time-independent Schrodinger equation solutions for an infinite well from 0 to a are of the form: \psi_n(x) = \sqrt{2/a} \sin (n \pi x/ a) If you move the well over so it is now from -a/2 to a/2, then you can replace x with x-a/2 and get the new equations right? If I...
  9. M

    Proving Hv = 0 for a Symmetric Matrice with Orthogonal Diagonalization

    Homework Statement Suppose H is an n by n real symmetric matrix. v is a real column n-vector and H^(k+1)v = 0. Prove that Hv = 0 The Attempt at a Solution Since H is a real symmetric matrice we can find an orthogonal matrix Q to diagnolize it: M = Q transpose. MA^(k+1)Qv = 0...
  10. W

    Symmetric Matrices as Submfld. of M_n. Prelim

    Hi, everyone. I am preparing for a prelim. in Diff. Geometry, and here is a question I have not been able to figure out: I am trying to show that Sym(n) , the set of all symmetric matrices in M_n = all nxn matrices, is a mfld. under inclusion. I see two...
  11. J

    Symmetric Potential: Reasons for Eigenstate Solutions

    I never learned this in the lectures (maybe I was sleeping), but now I think I finally realized what is the reason that eigenstate solutions of SE with a symmetric potential are either symmetric or antisymmetric. Is the argument this: "The Hamiltonian and the space reflection operator...
  12. Loren Booda

    Can a spherically symmetric antenna radiate?

    Seems to me I was taught in college physics that either a spherical "antenna" could not radiate or an antenna could not radiate spherically. Are either true? How about for an acoustical spherical membrane? For quadrupole mediated gravity?
  13. P

    A short one on symmetric matrices

    This isn't really homework, but close enough. I suppose this is quite simple, but my head's all tangled up for today. Anyways, Given the real symmetric matrix LTL = UDUT, find L. I suppose L = +- D1/2UT, and it's clear this choice of L satisfies the given equation. But can it be proven that...
  14. S

    Statics problem: finding forces on symmetric supports due to beam

    Homework Statement I only just started thinking about this, so I apologize if I can't frame it correctly... But say I have a completely uniform beam sitting on, say, 8 supports all distributed evenly about the beam's center of mass (which is also its geometric center). That is, for every one...
  15. F

    How Do You Prove Matrices Like AA^T and A+A^T Are Symmetric?

    I'm having trouble understanding a certain matrix problem. -Show that AA^T and A^TA are symmetric. -Show that A+A^T is symmetric. Any help would be greatly appreciated.
  16. B

    Is a Second Order Symmetric Tensor Always Represented by a Symmetric Matrix?

    Is the matrix of a second order symmetric tensor always symmetric (ie. expressed in any coordinate system, and in any basis of the coordinate system)? Please help! :blushing: ~Bee
  17. D

    Binary symmetric channel capacity

    Hi to our nice community. I want to learn why in a binary symetric channel the channel is calculated as C=1+plogp+(1-p)log(1-p) I only know that the channel is denoted as C=maxI(X;Y) btw what ; means in X;Y? Unfortunately my book doesn't mention these things so if u can reply me or...
  18. T

    Symmetric matrices and orthogonal projections

    Homework Statement Consider a symmetric n x n matrix A with A² = A. Is the linear transformation T(x) = Ax necessarily the orthogonal projection onto a subspace IR^n? Homework Equations The Attempt at a Solution No idea what thought to begin with.
  19. L

    Irrotational field -> Symmetric Jacobian

    Does anyone know any reference or proof to the statement that since a flow is irrotational, the Jacobian is symmetric?
  20. S

    Angular momentum of a particle in a spherically symmetric potential

    Homework Statement A particle in a spherically symmetric potential is in a state described by the wavepacked \psi (x,y,z) = C (xy+yz+zx)e^{-alpha r^2} What is the probability that a measurement of the square of the angular mometum yields zero? What is the probability that it yields...
  21. E

    Symmetric matrix and diagonalization

    This is a T/F question: all symmetric matrices are diagonalizable. I want to say no, but I do not know how exactly to show that... all I know is that to be diagonalizable, matrix should have enough eigenvectors, but does multiplicity of eigenvalues matter, i.e. can I say that if eignvalue...
  22. A

    How to test if a distribution is symmetric?

    How to test if a distribution is symmetric?? Hi all: To test if a distribution is symmetric or not, I knew we can use the mean-median == 0 and skewness == 0 I am wondering if there is any other methods of doing so? Also, which one of them are more...
  23. J

    Cylindrically symmetric current distribution: Magnetic field in all space

    Homework Statement a. An infinite cyclindrically symmetric current distribution has the form \vec J (r, \phi, z) = J_0 r^2/R^2 \ \ \ \vec\hat \phi for R<r<2R. Outside the interval, the current is 0. What is the field everywhere in space? b. An infinite cyclindrically symmetric current...
  24. R

    Finding a Basis for 3x3 Symmetric Matrices

    This is the problem that I am working on. Find a basis for the vector space of all 3x3 symetric matricies. Is this a good place to start 111 110 100 using that upper triangular then spliting it into the set. 100 010 001 000 000 000 000 000 000 100 010 000 000 000 000...
  25. M

    Even parity => symmetric space wave function?

    If I have af wavefunction that is a product of many particle wavefunctions $\Psi = \psi_1(r_1)\psi_2(r_2) ... \psi_n(r_n)$ If I then know that the parity of $ \Psi $ is even. Can I then show that the wavefunction i symmetric under switching any two particles with each other. That is...
  26. P

    Proving Boundedness of Symmetric Operator on Hilbert Space

    Homework Statement Let A be a linear operator on a Hilbert space X. Suppose that D(A) = X, and that (Ax, y) = (x, Ay) for all x, y in H. Show that A is bounded. The Attempt at a Solution I've tried to prove it by using the fact that if A is continuous at a point x implies that A is...
  27. R

    Chapter07.pdfCan Killing Vectors Derive the Schwarzschild Metric?

    "Is it possible to derive the Schwarzschild metric from Killing vectors, thus saving all that work with the Ricci tensor etc."
  28. B

    Integrating x^2tan x+y^3+4 over Area Symmetric

    Homework Statement Help! I was wondering if anyone can help me integrate: ∫ x^2tan x + y^3 +4 dA, where D is the region represented by D = {(x,y)|x2+y2≤2} Homework Equations I think that the area is symmetric, and so basically you only need to evalute from 0≤ x ≤ √(2-y^2) and 0≤ y ≤...
  29. P

    Calculating Theta in a Rotating Symmetric System

    Homework Statement A figure is attached. The symmetric system consists of 4 massless rodes with length 'L',the purple thing is a spring(the spring constant is k),when the system is at rest theta=45 deg. the system starts to rotate with angular speed 'w',the rodes are pushing on the...
  30. S

    Bound states for a Spherically Symmetric Schrodinger equation

    Homework Statement A particle of mass m moves in three dimensions in a potential energy field V(r) = -V0 r< R 0 if r> R where r is the distance from the origin. Its eigenfunctions psi(r) are governed by \frac{\hbar^2}{2m} \nabla^2 \psi + V(r) \psi = E \psi ALL in spherical coords...
  31. S

    Do Any Spacetimes with 7 to 9 Killing Vectors Exist?

    In 4 dimensions, a spacetime can have a maximum of 10 linearly independent Killing vectors. Are there known examples of spacetimes (satisfying Einstein's equation) with 7, 8, or 9 Killing vectors? I know FRW cosmologies have 6 Killing vectors, but I'm looking for something a bit more symmetric...
  32. M

    Form factor - spherically symmetric

    1) Use the fact that the form factor, F(q), is the Fourier transform of the normalised charge distribution p(r), which in the spherically symmetric case gives , F(q) = \int \frac{4\pi\hbar r}{q}p(r)sin(\frac{qr}{\hbar}) dr to find an expression for F(q) for a simple model of the proton...
  33. J

    Symmetric Difference Qutotient

    Show agrebraically that symmetric difference quotient produces the exact derivative f'(x) = 2ax+b for the quadractic function f(x) = ax^2+bx+c i know that: f(a + h) - f(a - h) Symmetric Difference Quotient = -------------------...
  34. E

    Commutator of a density matrix and a real symmetric matix

    Let p1,p2 be two density matrices and M be a real, symmetric matrix. Now, <<p1|[M,p2]>>= <<p1|M*p2>>-<<p1|p2*M>>= Tr{p1*M*p2}-Tr{p1*p2*M}= 2i*Tr{(Im(p1|M*p2))}. Why is it that this works out as simply as (x+iy)-(x-iy)? How is Tr{p1*p2*m}=conjugate(Tr{p1*M*p2})? I can't seem to figure...
  35. J

    About Singular and Symmetric Matrix

    I would like to know the statement is always true or sometimes false, and what is the reason: A is a square matrix P/S: I denote transpose A as A^T 1)If AA^T is singular, then so is A; 2)If A^2 is symmetric, then so is A.
  36. I

    Basis of set of skew symmetric nxn matrices

    Hi, I am having trouble with the question above. In general, I have trouble with questions like: What is the basis for all nxn matrices with trace 0? What is the dimension? What is the basis of all upper triangular nxn matrices? What is the dimension? Please help!
  37. H

    Is the Limit at x=0 of a Reflected Symmetric Function Negative Infinity?

    Lets say you have a function that is constant except at interval [-2,2] where it drains down to infinity. The whole function is reflected symetrically at x=0. Is the limit of this function, as x approaches 0, negative infinity?
  38. garrett

    Explore Geometry of Symmetric Spaces & Lie Groups on PF

    A few friends have expressed an interest in exploring the geometry of symmetric spaces and Lie groups as they appear in several approaches to describing our universe. Rather than do this over email, I've decided to bring the discussion to PF, where we may draw from the combined wisdom of its...
  39. P

    How do we know that the expansion of the universe is radially symmetric?

    Forgive me if I have used the wrong phrase to characterize the phenomenon. If my understanding is flawed, someone please correct me. From what I understand, theory postulates that all points in space will measure a red shift. How has this been tested? It seems to me such an effect would be...
  40. S

    Proving Nilpotency and Nonnegativity of Eigenvalues of Symmetric Matrices

    Show that every eigenvalue of A is zero iff A is nilpotent (A^k = 0 for k>=1) i m having trouble with going from right to left (left to right i got) we know that det A = product of the eignevalues = 0 when we solve for the eigenvalues and put hte characteristic polynomial = 0 then det...
  41. S

    [Discrete Math] Relations, symmetric and transitive

    Ok so here's one of the questions we've been assigned... So I can graphically see what this relation looks like, and from that I've shown it's reflexive. Now I'm working on proving it as being symmetric, but I can't put it into words. b) ~ is symmetric. Well we want to show that aRb ->...
  42. B

    How Is Symmetric Algebra Isomorphic to a Free Commutative R-Algebra?

    I have this problem that i need to prove and i don't even know where to start. So I have to show that the symmetric algebra ( Sym V ) is isomorphic to free commutative R-algebra on the set {x1, ..., xn}. Now i know that Sym V could be regarded as +Sym^n V for n>=0. And then we need to...
  43. T

    Conjugate Elements of a Symmetric Group

    Is the the following definition correct? Two elements a and b of a group G are said to be CONJUGATE if there exists g in G such that a=gbg^{-1}. For instance, show that all elements in the symmetric group S5 of order 6 conjugate.
  44. J

    Can someone help me about skew symmetric?

    Let A be an nxn skew symmetric mx.(A^T=-A). i) Show that if X is a vector in R^n then (X,AX)=0 ii) Show that 0 is the only possible eigenvalue of A iii)Show that A^2 is symmetric iv)Show that every eigenvalue of A^2 is nonpositive. v)Show that if X is an eigenvector of A^2 , then so is AX...
  45. CarlB

    PT symmetric non Hermitian formulation of QM

    I haven't read this yet, but I'm putting it up here for discussion as it seems so fascinating: PT-Symmetric Versus Hermitian Formulations of Quantum Mechanics Carl M. Bender, Jun-Hua Chen, Kimball A. Milton A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by...
  46. P

    Discriminant is a symmetric polynomial

    I've got to proof the following: Let f be a monic polynomial in Q[X] with deg(f) = n different complex zeroes. Show that the sign of the discriminant of f is equal to (-1)^s, with 2s the number of non real zeroes of f. I know the statement makes sense, because the discriminant is a...
  47. A

    Symmetric Matrices and Manifolds Answer Guide

    (1) If A is an n x n matrix, then prove that (A^T)A (i.e., A transpose multiplied by A) is symmetric. (2) Let S be the set of symmetric n x n matrices. Prove that S is a subspace of M, the set of all n x n matrices. (3) What is the dimension of S? (4) Let the function f : M-->S be defined by...
  48. P

    Ground State Symmetry of Single Electron w/ Non-Interacting 2nd Electron

    for a symmetrical potential with one electron, i know that the wavefunctions are symmetric or antisymmetric. for the ground state why is the wavefunction symmetric? Also, if you add a second electron that is non-interacting, (why) does it have the same wavefunction as the first electron?
  49. P

    Proving Reflections in R^n are Symmetric Matrices

    done Hi there, I am have trouble with a proof. I have some steps done, but I am not sure if I am aproaching this correctly. the question is: Show that the matrix of any reflection in R^n is a symmetric matrix. I know that F(x) = Ax + b is an isometry where A is an Orthoganol matrix...
  50. C

    Prove that that every symmetric real matrix is diagonalizable?

    How can I prove that that every symmetric real matrix is diagonalizable? Thanks in advance
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