What is Symmetric: Definition and 563 Discussions

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. R

    Symmetric positive definite matrix

    Homework Statement In a symmetric positive definite matrix, why does max{|aij|}=max{|aii|} Homework Equations |aij|≤(aii+ajj)/2 The Attempt at a Solution max{|aij|}≤max{(aii+ajj)/2 max{|aij|}≤max{aii/2}+max{ajj/2} max{|aij|}≤\frac{1}{2}max{aii}+\frac{1}{2}max{ajj} then I...
  2. Fernando Revilla

    MHB Symmetric Graphs: f(x)=3^x and g(x)=(1/3)^x Explained

    I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.
  3. andyrk

    Center of Mass of Uniform Symmetric Bodies

    Why are centre of mass of all uniform symmetric bodies at there geometric centre? We know the following result: "Consider a system of point masses m1,m2,m3... located at co-ordinates (x1,y1,z1), (x2,y2,z2)...respectively. The centre of mass of this system of masses is a point whose co-ordinates...
  4. F

    Are second derivative symmetric in a Riemannian manifold?

    Hi all! I was wondering if \partial_1\partial_2f=\partial_2\partial_1f in a Riemannian manifold (Schwartz's - or Clairaut's - theorem). Example: consider a metric given by the line element ds^2=-dt^2+\ell_1^2dx^2+\ell_2^2dy^2+\ell_3^2dz^2 can we assume that...
  5. tom.stoer

    Spherical symmetric collapse of pressureless dust

    Is there an exact solution for the spherical symmetric collaps of pressureless dust? Can one see a Schwarzschild solution for r > Rdust with shrinking Rdust(t) ?
  6. phosgene

    Normalization of a symmetric wavefunction

    Homework Statement I need to find the normalization constant N_{S} of a symmetric wavefunction ψ(x_{1},x_{2}) = N_{S}[ψ_{a}(x_{1})ψ_{b}(x_{2}) + ψ_{a}(x_{2})ψ_{b}(x_{1})] assuming that the normalization of the individual wavefunctions ψ_{a}(x_{1})ψ_{b}(x_{2}), ψ_{a}(x_{2})ψ_{b}(x_{1}) are...
  7. T

    Symmetric Bilinear: What a Drag

    I am having problems showing the following: ##f## and ##g## are two linearly independent functions in ##E## and ##\theta : \mathbb{R} \to \mathbb{R}## is an additive map such that ##\theta(\mu \nu) = \theta(\mu)\nu +\mu \theta(\nu); \mu,\nu \in \mathbb{R}##. Show that the function; ##\psi...
  8. D

    How can I diagonalize this symmetric matrix?

    \begin{array}{ccc} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{array}
  9. M

    Spherically symmetric spacetime

    I know from classical physics that, for example, an electric field is spherically symmetric if its magnitude depends only on the distance r to the origin (and not on the angles \phi, \theta) and it's in radially inward or outward direction. But, what does it mean when spacetime is spherically...
  10. G

    Solve Invertible Skew Symmetric Matrix: Hints & Tips

    I am asking for some hints to solve this excercise. Given an invertible skew symmetric matrix $A$, then show that there are invertible matrices $ R, R^T$ such that $R^T A R = \begin{pmatrix} 0 & Id \\ -Id & 0 \end{pmatrix}$, meaning that this is a block matrix that has the identity matrix in two...
  11. H

    Expressing a symmetric matrix in terms on eigenvalues/vectors

    Homework Statement Generate a random 10 x 10 symmetric matrix A (already done in MATLAB) . Express A in the formHomework Equations ## A = ## \displaystyle \sum_{j=1}^{10}λ_j(A)v_jv^T_j\ for some real vectors ##v_j, j = 1, 2, . . . , 10.## The Attempt at a Solution I'm pretty sure the...
  12. M

    Eigen values and Eigenvectors for a special case of a symmetric matrix

    Hey guys if i have a vector x=[x1,x2, ... xn] what are the eigenvectors and eigenvalues of X^T*X ? I know that i get a n by n symmetric matrix with it's diagonal entries in the form of Ʃ xii^2 for i=1,2,3,. . . ,n Thank you in advance once again!
  13. M

    Find the symmetric matrix from eigen vectors

    Hello to all of you, Is there a way to get the matrix A=[a b c d] from the eigenvectors (orthogonal) matrix H= sin(x) cos(x) cos(x) -sin(x) or to pose it differently to find a matrix that has these 2 eigenvectors ? Thank you in advance . Michael
  14. N

    Symmetric Converging-Diverging Nozzle Areas?

    Hey guys I was wondering if a symmetric C-D nozzle has the same area at the inlet and exit, can the mach numbers at both points be the same? This situation would involve a normal shock which would be near the exit as well. I would assume the Mach number at the throat would be 1. With that...
  15. P

    Some general formulae for circular orbits in symmetric spacetimes

    Consider the equatorial plane of a spherically symmetric space-time. Then we can write the metric in the equatorial plane (theta=pi/2) in terms of three coordinates - [t, r, phi] ds^2 = -f(r) dt^2 + g(r) dr^2 + h(r) dphi^2 For the Schwarzschild metric we can write: f = c^2(1 -\frac{2 G...
  16. I

    Form of symmetric matrix of rank one

    Homework Statement The question is: Let C be a symmetric matrix of rank one. Prove that C must have the form C=aww^T, where a is a scalar and w is a vector of norm one. Homework Equations n/a The Attempt at a Solution I think we can easily prove that if C has the form...
  17. I

    MHB Form of symmetric matrix of rank one

    The question is:Let $C$ be a symmetric matrix of rank one. Prove that $C$ must have the form $C=aww^T$, where $a$ is a scalar and $w$ is a vector of norm one.(I think we can easily prove that if $C$ has the form $C=aww^T$, then $C$ is symmetric and of rank one. But what about the opposite...
  18. icesalmon

    Symmetric arc length of ln(x) and e^x

    Homework Statement Explain why ∫(1+(1/x2)1/2dx over [1,e] = ∫(1+e2x)1/2dx over [0,1] The Attempt at a Solution The two original functions are ln(x) and ex and are both symmetrical about the line y = x. If I take either of the functions and translate it over the line y = x the two...
  19. H

    Divergence and Radially Symmetric Fields

    Is it possible for a spherically symmetric field, on all of R^3, to have a divergence of 0? (assuming the field is nonzero) Relevant equation: F=f(ρ)a (a is a unit vector of <x,y,z>) and f(ρ) is scalar fxn, and ρ = lal
  20. H

    F(x,y,z) Symmetry in Scalar Function g: Counterexample or Confirmation?

    Today I've tried to investigate properties of a function f(x_1,x_2,x_3) satisfying \nabla\times(f\mathbf{x})=0 in \mathbb{R}^3, where some degrees of the differentiability are assumed if needed. By some basic procedures, I've deduced that: there is a scalar function g such that...
  21. Fernando Revilla

    MHB Quirino27's question at Yahoo Answers (R symmetric implies R^2 symmetric)

    Here is the question: Here is a link to the question: Prove that if a relation R on a set A is symmetric, then the relation R² is also symmetric? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  22. NATURE.M

    Find the vector, parametric, symmetric equations of a line

    Homework Statement Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°. L1: x = 4 + 2t y = 8 + 3t z = -1 − 4t L2: x = 7 - 6t y = 2+ t z = -1 + 2t Homework Equations vector, parametric, symmetric equations of line...
  23. NATURE.M

    Write parametric and symmetric equation for the z-axis

    Homework Statement Write parametric and symmetric equations for the z-axis. Homework Equations vector, parametric and symmetric equations, in general form. The Attempt at a Solution I believe I have obtained the correct answer, would just like confirmation. Let our direction...
  24. M

    Is this method for exchanging symmetric key using RSA sound?

    Bob know's Alice's public key, and he wants to make sure he's connecting to the one which has that key. Furthermore, Alice wants to verify when she gets a connection from Bob who'll give his public key that he is indeed the one who has that key. Bob will send Alice half the AES private key...
  25. C

    Antisymmetric and symmetric part of a general tensor

    I've seen it stated several times that a general covariant or contravariant tensor of rank n can be separated into it's symmetic and antisymmetric parts T^{\mu_1 \ldots \mu_n} = T^{[\mu_1 \ldots \mu_n]} + T^{(\mu_1 \ldots \mu_n)} and this is easy to prove for the case n=2, but I don't see how...
  26. S

    Proving that the free particle lagrangian is rotationally symmetric

    Homework Statement Show that the free particle lagrangian is invariant to rotations in $$\Re^{3}$$, but I assume this means invariant up to a gauge term. $$L=m/2 [\dot{R^{2}} + R^{2}\dot{θ^{2}} +R^{2}Sin^{2}(θ)\dot{\phi^{2}}$$ Homework Equations I consider an aribtrary infinitesimal...
  27. A

    Why are there multiple Slater determinants for the same number of electrons?

    For multibosonic systems, as I understand, the wave function must always be symmetric (antisymmetric for fermionic, which this question easily generalizes to). But as far as I can see for N>2 you can easily construct a lot of other wave functions which are symmetric rather than the one my book...
  28. C

    How Does Moving a Charge Affect Electric Flux in a Cubic Gaussian Surface?

    Homework Statement A point charge of negative polarity is located at the centre of a cubic Gaussian surface with edges of length ##0.5m##. Calculate the electric flux through one of the faces of the surface. What would happen if the charge was moved 10cm to the right? Homework Equations...
  29. T

    Permutations of a single number in the symmetric group

    Say we have the symmetric group S_5. The permutations of \{2,5\} are the identity e and the transposition (25). But what are all the permutations of \{3\}? Is it e and the 1-cycle (3)?
  30. matqkks

    MHB Repeated eigenvalues of a symmetric matrix

    I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?
  31. matqkks

    Repeated eigenvalues of a symmetric matrix

    I have been trying to prove the following result: If A is real symmetric matrix with an eigenvalue lambda of multiplicity m then lambda has m linearly independent e.vectors. Is there a simple proof of this result?
  32. S

    Eigenvalues of a complex symmetric matrix

    Eigen values of a complex symmetric matrix which is NOT a hermitian are not always real. I want to formulate conditions for which eigen values of a complex symmetric matrix (which is not hermitian) are real.
  33. N

    Symmetric difference of relationships

    Homework Statement 2. Let R1 = {(1,1),(1,2),(2,3),(3.4), (2,4) } and R2 = {(1,1),(2,2),(2,3),(3,3),(3,4) } be relations from {1,2,3} to {1,2,3,4} R1⨁ R2 Homework Equations The Attempt at a Solution R1⨁ R2 = {(1,2), (2,2), (2, 4), (3,3)} Is this correct?
  34. N

    Relationship: reflexive, symmetric, antisymmetric, transitive

    Homework Statement Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive. The relation R on P = {a, b, c} where R = {(a, a), (a, b), (a, c), (b, c), (c, b)} Homework Equations The Attempt at a Solution Not reflexive because there is...
  35. N

    Relationship: reflexive, symmetric, antisymmetric, transitive

    Homework Statement Determine which binary relations are true, reflexive, symmetric, antisymmetric, and/or transitive. The relation R on all integers where aRy is |a-b|<=3Homework Equations The Attempt at a Solution The relationship is reflexive because any number minus itself will be zero...
  36. S

    MAxwell reciprocal theorem and symmetric stiffness matrix

    As per Maxwell reciprocal theorem, it is valid only for elastic materials and structures indergoing small displacements.That is k12 = k21, kij = kji hence stiffness matrix is symmetric. Howbver, I just have been going through MY OWN written programs for geometric non linear problems and I...
  37. Z

    Vacuum solution with static, spherical symmetric spacetime

    Homework Statement I am trying to derive the line element for this geometry. But I am not sure how to show that ds can't contain any crossterms of d\theta and d\phi Homework Equations ds must be invariant under reflections \theta \rightarrow \theta'=\pi - \theta and \phi...
  38. D

    MHB Incorporate axis symmetric case

    With this code, I can't look at the $\mathcal{J}_0$ or $\mathcal{Y}_0$. How can I altered to to pick up those orders? ClearAll["Global'*"]; inits = Table[ FindRoot[ BesselJ[n, x]*BesselY[n, 2*x] - BesselJ[n, 2*x]*BesselY[n, x] == 0, {x, n + 3}], {n, 1, 5}]; g1 = x /. inits zeros...
  39. O

    How to write basis for symmetric nxn matrices

    Homework Statement Write down a basis for the space of nxn symmetric matrices. The Attempt at a Solution I just need to know what the notation for this sort of thing is. I understand what the basis looks like, and I was even able to calculate that it would have dimension...
  40. M

    Questions on the symmetric group

    first , if p is prime , show that an element has order p in Sn iff it's cycle decomosition is a product of commuting p-cycles my solution is very diffrent about the one in the book and I don't know if my strategy is right my proof ______ let T is an element of Sn and the cycle...
  41. T

    Defining symmetric mappings of R^3

    Consider the basis β = {(1,1,0), (1,0,-1) , (2,1,0)} for R3. Which of the following matrices A = [T]ββ (where T is the transformation) define symmetric mappings of R3? Attempt/ Issue: The properties that I know that define a matrix being symmetric are that <T(X), Y> = <X,T(Y)> i.e the innner...
  42. B

    Symmetric Difference Proof for Sets A, B, and C

    Homework Statement Let % be the symmetric difference. Prove that for any sets A, B, C; A%B=C iff B%C=A iff A%C=B Homework Equations (I will use forwardslash as I cannot find the backslash on my keyboard.) The Attempt at a Solution Take x in A%B. Then x is either in A/B or in...
  43. J

    Variational Principle of 3D symmetric harmonic oscillator

    Homework Statement Use the following trial function: \Psi=e^{-(\alpha)r} to estimate the ground state energy of the central potential: V(r)=(\frac{1}{2})m(\omega^{2})r^{2} The Attempt at a Solution Normalizing the trial wave function (separating the radial and spherical part)...
  44. S

    Square wave symmetric around zero volts

    Hi everyone For a pre-lab, I am asked to draw a square wave symmetric around zero volts. I am not sure what this graph looks like, can someone give me an example? Thank You
  45. S

    Spherically symmetric charge density given electric potential

    1. A spherically symmetric charge distribution results in an electric potential of the form What is the charge distribution? 2. Hint: consider the difference in electric field between two values of r Show that the answer is of the form 3. I have attempted several solutions but haven't...
  46. O

    MHB Finding the Inverse of Symmetric Matrices with Non-Real Coefficients

    Hello everyone! I'm struggling to find a general formula for obtaining an inverse of a symmetric matrix, for e.g. 1 i -1 i -i 2 -1 2 1 Any help is appreciated!
  47. C

    Eigenvectors of a symmetric matrix.

    Is it true that an nxn symmetric matrix has n linearly independent eigenvectors even for non-distinct eigenvalues? How can we show it rigorously? Basically, I want to prove that if an nxn symmetric matrix has eigenvalue 0 with multiplicity k, then its rank is (n - k). If we can prove that there...
  48. G

    Sphereically Symmetric potential well

    Homework Statement A particle that movies in three dimensions is trapped in a deep spherically symmetric potential V(r): V(r) = 0 at r < r_{}0 --> ∞ at r ≥ r_{}0 where r_{}0 is a positive constant. The ground state wave function is spherically symmetric, so the radial wave function u(r)...
  49. I

    Symmetric positive definite matrix

    Homework Statement (i) Show that if A is symmetric positive semi-definite, then there exists a symmetric matrix B such that A=B^2. (ii) Let A be symmetric positive definite. Find a matrix B such that A=B^2.Homework Equations The Attempt at a Solution For part 1, I used: B = Q\sqrt{\Lambda}...
  50. C

    Relations, Set Theory, Reflexive, Symmetric, Transitive

    Homework Statement Determine whether the relations on three sets are Reflexive, Irrelfexive, Symmetric,, Asymmetric, Antisymmetric, Transitive, and Intransitive. The relation \subseteq on a set of sets. Homework Equations The Attempt at a Solution I am having trouble figuring out...
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