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

    Difference symmetric matrices vector space and hermitian over R

    Hi guys, I have a bit of a strange problem. I had to prove that the space of symmetric matrices is a vector space. That's easy enough, I considered all nxn matrices vector spaces and showed that symmetric matrices are a subspace. (through proving sums and scalars) However, then I was asked...
  2. L

    Is there a way to diagonalize a symmetric matrix without using a calculator?

    Homework Statement I need to diagonalize the matrix A= 1 2 3 2 5 7 3 7 11 The Attempt at a Solution Subtracting λI and taking the determinant, the characteristic polynomial is λ3 - 17λ2 + 9λ - 1 (I have checked this over and over) The problem now is it has some ugly roots, none that I would...
  3. C

    Symmetric and Antisymmetric WF

    Hello, Why do symmetric wave function has less energy than the anti symmetric wave function and how does it connect to the number of the nodes (why existence of a node point in the anti symmetric tells us that this is more energetic function?)
  4. H

    Subsets of symmetric groups Sn

    This is not a homework question, just a question that popped into my head over the weekend. My apologies if this is silly, but would you say that the symmetric group S4 is a subset of S5? My friends and I are having a debate about this. One argument by analogy is that we consider the set...
  5. I

    MATLAB [Matlab]Copy Lower Triangle of symmetric matrix to Upper Triangle(or visa versa)

    Hello all! I just had a question about combining elements of matrices. In the MATLAB documentation, there was a function called triu and tril that extracts the upper and lower components of a matrix, respectively. I was wondering if there was a way to copy the elements of the upper triangle...
  6. B

    When the gradient of a vector field is symmetric?

    Homework Statement "A gradient of a vector field is symmetric if and only if this vector field is a gradient of a function" Pure Strain Deformations of Surfaces Marek L. Szwabowicz J Elasticity (2008) 92:255–275 DOI 10.1007/s10659-008-9161-5 f=5x^3+3xy-15y^3 So the gradient of this function...
  7. T

    Discrete Math: Symmetric Closure & Numerical Analysis

    Discrete Mathematics -- Symmetric Closure Math help in Numerical Analysis, Systems of I can't seem to find the way to approach this problem. Because it has symbols I don't know how to type here, I have attached an image here instead. Please help me if you can. Any input would be greatly...
  8. T

    Solving Symmetric Equations to Determine if Points Lie on Line L

    Homework Statement My question is how do you use the symmetric equation. For instance I have a question that states: A line L has parametric equations x=4+3t, y=3+4t, z=9-4t. Determine whether or not the points given lie on the line L. points (17, 14, -9). Homework Equations I know that...
  9. P

    Proof of symmetric and anti symmetric matrices

    Homework Statement aij is a symmetric matrix bij is a an anti symmetric matrix prove that aij * bij = 0 Homework Equations aij * bij The Attempt at a Solution any one got any ideas ?
  10. M

    Parametric equations and symmetric equations

    Homework Statement Find parametric equations and symmetric equations for the line through the points (0,1/2,1) and (2,1,-3) Homework Equations The Attempt at a Solution I started out graphing the points and connecting them with a straight line. I called the first point P...
  11. B

    Classifying Symmetric Quadratic Forms

    Hi, All: I am trying to see how to classify all symmetric bilinear forms B on R^3 as a V.Space over the reals. My idea is to use the standard basis for R^3 , then use the matrix representation M =x^T.M.y . Then, since M is, by assumption, symmetric, we can diagonalize M...
  12. F

    Integral of the reflection operator in arbitrary symmetric spaces.

    Just as the title says, suppose X is a symmetric manifold and \hat{S}(x) is the linear operator associated to \sigma_x\in G for some unitary irreducible representation, where \sigma_x is the group element that performs reflections around x (remember X=G/H for H\subset G). Now take the...
  13. C

    Relation which is reflexive only and not transitive or symmetric.

    Homework Statement Relation which is reflexive only and not transitive or symmetric? Homework Equations No equations just definitions. The Attempt at a Solution I can find a relation for the other combinations of these 3 however, I cannot find one for this particular combination...
  14. Shackleford

    Finding Distinct Elements of G/H in Symmetric Group S4

    http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110804_134032.jpg?t=1312484230 http://i111.photobucket.com/albums/n149/camarolt4z28/IMG_20110804_134038.jpg?t=1312484242 I found that the order of G/H is 6. According to the Lagrange's Thereom, order of G = order of H *...
  15. P

    Symmetric Difference if/then Proof

    Hi there, I'm trying to figure out proving the following: if X oplus Y = Y oplus X then X = Y In order to prove it, I need to use the symmetric difference associativity & other characteristics and identities. Can you please give me a direction? Please explain the answer as a teacher...
  16. P

    Eigenvalues of a symmetric operator

    I'm reading from Wikipedia: I thought linear operators always had eigenvalues, since you could always form a characteristic equation for the corresponding matrix and solve it? Is that not the case? Are there linear operators that don't have eigenvalues?
  17. U

    Symmetric difference in sets

    Homework Statement There is a symmetric difference in sets X & Y, X Y is defined to be the sets of elements that are either X or Y but not both Prove that for any sets X,Y & Z that (X\oplusY)\oplus(Y\oplusZ) = X\oplusZ Homework Equations \oplus = symmetic difference The Attempt at...
  18. F

    Time Symmetric Interpretation

    Why are the Time Symmetric Interpretation rarely if ever brought up in discussions here? It restores determinism and realism. This article explains the jist of the interpretation and experimental evidence...
  19. H

    Sumation of symmetric and skew symmetri metrices

    Express \left(\begin{array}{cccc} 6 & 1 & 5\\ -2 & -5 & 4\\ -3 & 3 & -1\ end{array} \right) as the sum of the symmetric and skew symmetric matrices. I did this following way Consider symmetric metric as "A" then; A = \left(\begin{array}{cccc} 6 & 1 & 5\\ 1 & -5 & 4\\ 5 & 4 & -1\ \end{array}...
  20. F

    Symmetric looking equations needing a symmetric solution

    Hi all, I have a set of equations that look very nice and symmetric, but the only way I'm able to find solutions to them is with pages and pages of algebra! Can any members with more of a mathematical flair than myself point me in the direction of a more direct and satisfactory method of...
  21. K

    Intermediate subgroups between symmetric groups

    Homework Statement For n>1, show that the subgroup H of S_n (the symmetric group on n-letters) consisting of permutations that fix 1 is isomorphic to S_{n-1} . Prove that there are no proper subgroups of S_n that properly contain H.The Attempt at a Solution The first part is fairly...
  22. S

    Prove the symmetric group is generated by

    Homework Statement Use induction of n to prove that the transpositions s_i = (i, i+1), 1 \leq i \leq n - 1 generate S_n. Homework Equations Notation: e = Identity permutation. Any permutation can be written as a disjoint product of transpositions. The Attempt at a Solution...
  23. T

    Get Lorentzian Spherically Symmetric Metric to Sylvester Form

    Hi, I'm trying to determine the exact transformation that brings a spherically symmetric spacetime metric in spherical coordinates to the Sylvester normal form (that is, with just 1 or -1 on its main diagonal, with all other elements equal to zero.) Assuming that the metric has Lorentzian...
  24. Y

    Diagonalization of symmetric bilinear function

    According to duality principle, a bilinear function \theta:V\times V \rightarrow R is equivalent to a linear mapping from V to its dual space V*, which can in turn be represented as a matrix T such that T(i,j)=\theta(\alpha_i,\alpha_j). And this matrix T is diagonalizable, i.e...
  25. J

    The Big Bounce and the Parity Problem

    Is it possible that whatever cause the big bang to happen and make space expand also (for lack of a better phrase) tore time in two? Resulting in two universes moving in opposite directions of time, and could this be used to explain why there appears to be more matter then antimatter in the...
  26. W

    Symmetric Part of a Mixed (1,1) Tensor

    I have read in a couple of places that mixed tensors cannot be decomposed into a sum of symmetric and antisymmetric parts. This doesn't make any sense to me because I thought a mixed (1,1) tensor was basically equivalent to a standard linear transform from basic linear algebra. I am also...
  27. S

    Symmetric Polynomial Explained for Homework

    Homework Statement No problem exactly I am just reading a book that refrences symmetric polynomials but i don't know what a symmetric polynomial is. I looked at the wiki page but i didn't really get what it was saying. Any help on clearing up the meaning would be greatly appreciated...
  28. S

    Potential of a cylindrically symmetric quadrupole

    Homework Statement The question is in the attached pdf but is also relatively what I have written in (2) below. Homework Equations My question is in two parts; (i) What does it mean, a 'cylindrically symmetric quadrupole? Is there a geometrical interpretation? (ii) How do we show that...
  29. R

    Quantizating a symmetric Dirac Lagrangian

    As is well known, a Dirac Lagrangian can be written in a symmetric form: L = i/2 (\bar\psi \gamma \partial (\psi) - \partial (\bar\psi) \gamma \psi ) - m \bar\psi \psi Let \psi and \psi^\dagger be independent fields. The corresponding canonical momenta are p = i/2 \psi^\dagger...
  30. H

    Symmetric bilinear forms on infinite dimensional spaces

    It is a well known fact that a symmetric bilinear form B on a finite-dimensional vector space V over any field F of characteristic not 2 is diagonalisable, i.e. there exists a basis \{e_i\} such that B(e_i,e_j)=0 for i\neq j. Does the same hold over an infinite dimensional vector space...
  31. K

    Assistance with symmetric group/special linear group

    thanks
  32. P

    Diagonalization of complex symmetric matrices

    Is every complex symmetric (NOT unitary) matrix M diagonalizable in the form U^T M U, where U is a unitary matrix? Why?
  33. N

    How do vector functions behave under transformations for symmetry?

    Hi, How does one define symmetry of a system? I believe that a scalar function g(\vec x) is called symmetric under a transformation \vec F(\vec x) if and only if g(\vec x) = g(\vec F(\vec x)) If there is an equivalent criteria for vector functions, I would be inclined to define a...
  34. S

    Bilinear Form & Linear Functional: Symmetric & Coercive?

    Homework Statement The bilinear form are symmetric, i.e. a(u,v) = a(v,u) for all u and v. Find the bilinear form and the linear functional for the problem -\Deltau + b . \nablau + cu = f(x) in \Omega u = 0 on the boundary. Is this bilinear form for this problem symmteric? Is it coersive...
  35. B

    Proving the Even Rank of Skew Symmetric Matrices: Induction and Other Methods

    how can we prove that the rank of skew symmetric matrix is even i could prove it by induction is there another way
  36. P

    Cholesky for complex *symmetric*

    Hi, I am working with a Galerkin FEM implementation of an elastodynamic problem in the frequency domain. For the purely elastic case, this results in a symmetric, positive definite linear system that is efficiently solved by Cholesky decomposition. In order to consider anelasticity, however...
  37. J

    Equations for sliding/rolling symmetric top

    Homework Statement I am attempting to derive the equations of motion for a sliding/rolling (either case or both cases) symmetric spinning top that rises under the influence of sliding/rolling friction. This is a 6 degree of freedom system with the 3 Euler angles and 3 xyz directions (although...
  38. D

    Time Symmetric Quantum Mechanics

    I'm just trying to get a feel for how seriously this theory is being considered these days. For those not familar with it, here's a somewhat okay laymans description: http://discovermagazine.com/2010/apr/01-back-from-the-future/article_view?searchterm=Tollaksen&b_start:int=0 Also...
  39. N

    Directed Graphs: Reflexive, Symmetric, Transitive

    Homework Statement Hello, I want to make sure that I graphed the directed graphs in my homework correctly. The problems and my work is located in the attachment. I also uploaded the directed graphs onto this link: http://img857.imageshack.us/f/83289329.png/" Homework Equations NoneThe Attempt...
  40. H

    Conclusions from Symmetric Equations Identity

    Homework Statement What conclusion can be drawn from the lines (x-x0)/a = (y-y0)/b = (z-z0)/c (x-x0)/A = (y-y0)/B = (z-z0)/C if aA + bB +cC = 0 Homework Equations The Attempt at a Solution I put everything in parametric form but that didn't do much for me. Is...
  41. R

    Find a basis for the space of 2x2 symmetric matrices

    a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis. b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.
  42. L

    Subgroup of a symmetric group Sn

    Homework Statement Show that if G is a subgroup of a symmetric group Sn, then either every element of G is an even permutation or else exactly half the elements of G are even permutations. Homework Equations The Attempt at a Solution We have a hint for the problem. If all the...
  43. B

    Calculating Representation of Linear Operator for Symmetric Matrix

    Homework Statement Let L(x) a linear operator defined by setting the diagonal elements of x to zero. What will be the representation of this operator to the following basis set? x E X. X denote the set of all real symmetric 3x3 matrices. Homework Equations L*y=x L=x*inv(y)...
  44. pellman

    Does the metric have to be symmetric? Why?

    Why must we have g_{\mu\nu}=g_{\nu\mu}? What are the physical consequences if this did not hold?
  45. M

    Mathematica Defining function of a vector and symbolic symmetric matrices- mathematica

    Hi all, I'd like to define a vector valued function in mathematica 7 as the exponential of a quadratic form, defined with respect to a purely symbolic matrix. What I want to do with it is to take derivatives with respect to the components of my vector, and evaluate the result when all...
  46. I

    Show that diagonal entries of a skew symmetric matrix are zero.

    I'm pretty inexperienced in proof writing. So not sure if this was valid. If a matrix is skew symmetric then A^T = - A, that is the transpose of A is equal to negative A. This implies that if A = a(i,j), then a(j,i) = -a(i,j). If we're referring to diagonal entries, we can say a(j,j) =...
  47. C

    Basis of skew symmetric matrix

    Homework Statement Let W be a 3x3 matrix where A^t(transpose)=-A. Find a basis for W. Homework Equations Find a basis for W. The Attempt at a Solution I have no idea how to start it.
  48. C

    Unravelling the Mystery of the (0,3) Symmetric Tensor

    I am a bit confused by this observation. Every tensor is it's symmetric plus antisymmetric part. Thus for the components of a (0,3) tensor F_{\lambda\mu\nu}=F_{[\lambda\mu\nu]}+F_{\{\lambda\mu\nu\}} and if I write this down explicitly I end up that for the components of ANY (0,3)...
  49. I

    Matrix representation of a closed symmetric operator

    Homework Statement Show that a closed symmetric operator has a matrix representation. Homework Equations There are lots. I'm hoping somebody familiar with linear operators in Hilbert spaces is reading this! The Attempt at a Solution Hi, I'm trying to prove that a closed...
  50. P

    Electrostatics - spherically symmetric charge density

    Homework Statement Imagine a spherically symmetric charge density p(x) = Cr for r≤a and 0 for r>a (a) determine the electric field E(x) and potential V(r). Notice that V(r) and E(x) are continuous at r=a. (b) Now suppose additional charge is placed uniformly on the surface at r=a, with...
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