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. 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)...
  2. 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}...
  3. 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...
  4. B

    Basic Set Theory: Determining Relations: Reflexive, Symmetric, Transitive

    I am taking a philosophy course that covers basic set theory as part of the introduction. I’m not sure in which section of the forum set theory should be, but I think this is the right place. Homework Statement For each of the following relations, indicate whether it is Reflexive...
  5. F

    Diagonalizing a symmetric matrix with non-distinct eigenvalues

    Homework Statement Given the matrix A: 4 2 2 2 4 2 2 2 4 Find the matrix P such that P-1AP is diagonal Homework Equations The Attempt at a Solution So I had this question today on a placement exam and it threw me for a loop. I found the eigenvalues to be 2,2, and 8. The...
  6. S

    Find symmetric equations for the line of intersection of the planes

    Homework Statement Find symmetric equations for the line of intersection of the planes The planes: 5x - 2y - 2z = 1 4x + y + z = 6 Homework Equations r = r0 + tv x = x0 + at y = y0 + bt z = z0 + ct The Attempt at a Solution I have attempted this in many different manners and would like...
  7. S

    Are These Relations Symmetric?

    Determine which of these relations are symmetric 1) x~y if and only if x-y is positive 2) x~y if and only if xy >= 0 3) x~y if and only if x+2y is positive 4) x~y if and only if x+y is positive 5) x~y if and only if x+y is odd I thought all but 1) but this was wrong. The only one I...
  8. T

    1. Symmetric difference; 2. Commutativity of natural numbers

    Homework Statement I have two problems that I got stuck. 1. \exists ! N\in P(X) , A\Lambda N=A, \forall A\in P(X) and for each A\in P(X), \exists ! A'\in P(X) , A\Lambda A' =N 2. Prove a+(b+c) = (a+b) +c, for positive integers a, b, cHomework Equations 1. Given sets A,B \in P(X), where P(X)...
  9. D

    Determine Reflexive, Transitive, Symmetric of R Relation

    Homework Statement Determine if the following relation is reflexive, transitive, symmetric or anti-symmetric. (A,B) element of R(relation) if for every epsilon > 0, there exists a element of A and b element of B with |a-b| < epsilon. Homework Equations The Attempt at a Solution I already...
  10. TrickyDicky

    Riemannian symmetric spaces and Lie algebras

    I'm interested in the crossover of Lie groups/differential geometry and I'm a bit confused about the relation of Lie algebras with symmetric spaces. Take for instance the Lie group G=SL(2,R), we take the quotient by K=SO(2) as isotropic group(maximal compact subgroup) and get the symmetric...
  11. V

    Metric of a static, spherically symmetric spacetime

    The (0,0) and (r,r) components are: g_{00}= -e^{2\phi},g_{rr}=e^{2\Lambda}. From the first component, combined with the fact that the dot product of the four velocity vector with itself is -1, one can find in the MCRF, U^0=e^{-\phi}. What does this mean? In the MCRF, the rate of the two clocks...
  12. S

    Well-defined map: transvections in symmetric space

    Hi! I'm trying to understand a proof for the fact that the isometry group of a symmetric space is a Lie group. The proof uses a lemma and I don't see how the lemma works. Here is the statement in question: (Let me give you the definition for \tau_v: Let M be a symmetric space and c:\mathbb R...
  13. T

    Time symmetric quantum mechanics

    I have been asking during the last couple of days about EPR, measurement problem and all those sort of things. As a consequence I arrived to this funny interpretation in which (as wikipedia says) locality, determinism and a lot of desirable (to me) properties are preserved. Nevertheless I saw in...
  14. T

    Spherically Symmetric charge distribution

    I am currently doing a past paper question for my electromagnetism exam and I can't seem to figure out this problem, it is probably quite simple but I can't see a solutionHomework Statement Consider a spherically symmetric charge distribution: ρ(r) = ρ0(r/r0)-n for r>r0 ρ(r) = ρ0 for r≤r0...
  15. W

    Generators and Defining Relations on the Symmetric Group of degree n

    Homework Statement I am working through MacLane/Birkhoff's Algebra, and in the section on Symmetric and Alternating groups, the last few exercises deal with generators and Defining relations for Sn (the symmetric group of degree n). These read: 11. Prove that Sn is generated by the cycles (1...
  16. T

    Induction on the n-dimensional, radially symmetric wave equation

    Homework Statement Consider the radially symmetric wave equation in n dimensions u_{tt} = u_{rr} + \frac{n-1}{r}u_r Use induction to show that the solution is u = \left(\frac{1}{r}\frac{\partial}{\partial r}\right)^{(n-3)/2} \frac{f(t-r)}{r} for n odd and u =...
  17. T

    Symmetric and Alternating Groups disjoint cycles

    Homework Statement Let a = (a1a2..ak) and b = (c1c2..ck) be disjoint cycles in Sn. Prove that ab = ba. The Attempt at a Solution Sn consists of the permutations of the elements of T where T = {1,2,3,...,n} so assume we take an i from T. Then either i is in a, i is in b, or i is in...
  18. M

    Gaussian Beam in a Symmetric Confocal Resonator.

    λHomework Statement A symmetric confocal resonator with mirror spacing d =16 cm, mirror reflectances 0.995, and n = 1 is used in a laser operating at λ[o] = 1 μm. (a) Find the radii of curvature of the mirrors. (b) Find the waist of the (0,0) (Gaussian) mode. (c) Sketch the intensity...
  19. T

    Symmetric matrices and Newton's third law

    So, I was studying coupled oscillations and came across a statement that I couldn't figure out. It was that a particular matrix was symmetrical by Newton's Third Law. I know what Newton's Third Law is, I know what symmetric matrix is. But, for example, a matrix like this: -2k/m...
  20. A

    Hartree Fock Symmetric Energy Expression

    Hello. I just wonder why the energy expression of Hartree Fock method is symmetric. I tried to find out the reason on the Internet but I could only find that: since the Hartree Fock energy expression is symmetric, it is variational. In Hartree Fock method, the repulsive energy between...
  21. E

    Proving one element in the symmetric group (s>=3) commutes with all element

    I am really stuck with how to prove that the only element in Sn (with n>=3) commuting with all the other elements of this group is the identity permutation id. I have no idea what I am supposed to do with it, i know why S3 has only one element that commutes but i don't know how to prove it...
  22. A

    Hartree Fock Method Symmetric Energy Expression

    Hello. I just wonder why the energy expression of Hartree Fock method is symmetric. I tried to find out the reason on the Internet but I could only find that: since the Hartree Fock energy expression is symmetric, it is variational. In Hartree Fock method, the repulsive energy between...
  23. G

    Determinant of symmetric matrix with non negative integer element

    Let \begin{equation*} A=% \begin{bmatrix} 0 & 1 & \cdots & n-1 & n \\ 1 & 0 & \cdots & n-2 & n-1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ n-1 & n-2 & \cdots & 0 & 1 \\ n& n-2 & \cdots & 1 & 0% \end{bmatrix}% \end{equation*}. How can you prove that det(A)=[(-1)^n][n][2^(n-1)]? Thanks.
  24. G

    Is the Determinant of a Symmetric Matrix with Zero Diagonal Elements Non-Zero?

    How to prove that the determinant of a symmetric matrix with the main diagonal elements zero and all other elements positive is not zero and different ?
  25. B

    Projectile Motion is Symmetric

    Okay, I read that in the case of no air-resistance, projectile motion is symmetric; that the initial velocity will equal the final velocity, in magnitude; and that a projectile traveling upwards, achieving a zero velocity of the vertical component, will have to fall the same horizontal distance...
  26. S

    Spherically Symmetric Charge Distribution

    Homework Statement Consider a spherically symmetric charge distribution \rho = \rho (r) Homework Equations By dividing the charge distribution into spherical shells, find the potential \phi and the electric field strength \bf{E} in terms of \rho (r) The Attempt at a Solution The...
  27. S

    Does diagonalizable imply symmetric?

    In order to prove my PDE system is well-posed, I need to show that if a matrix is diagonalizable and has only real eigenvalues, then it's symmetric. Homework Equations I've found theorems that relate orthogonally diagonalizable and symmetric matrices, but is that sufficient? The...
  28. H

    Symmetric vector to tensor representation?

    Hi all! I have a discrete 2D vector field with a particular characteristic: At every point, instead of having a single vector, I have two vectors which are in the opposite direction. For example, at point p(x,y)=p(0,0) I have two vectors: v1(1,1) and v2(-1,-1). And so on for all points. I...
  29. A

    Proving irreflexive and symmetric relation

    Homework Statement I am on the final part of a question and I have to prove that the following is a irreflexive symmetric relation over A or if it is not then give a counter example. R is given as an irreflexive symmetric relation over A. Relation: {(X, Y) | X ⊆ A ∧ Y ⊆ A ∧ ∀x ∈ X.∀y ∈...
  30. S

    Maximum final component of a simple symmetric multidimensional random walk?

    I am developing a simple probabilistic model of my own mistakes in (1) solving math problems and (2) implementing algorithms on a computer. I have reduced the problem to one which seems simple enough, but which I have been unable to solve, due to my mathematical inexperience. I figured I would...
  31. A

    Help needed finding vector, parametric, symmetric equation

    Homework Statement Find the vector, parametric and symmetric equations of a line that intersect both line 1 and line 2 at 90°. line 1: x = 4 + 2t y = 8 + 3t z = -1 - 4t line 2: x = 7 - 6t y = 2+ t z = -1 + 2t Homework Equations not sure. I am not asking for the answer...
  32. I

    Reflexive, Symmetric, Transitive

    Indicate if the following relation on the given set is reflexive, symmetric, transitive on a given set. R where (x,y)R(z,w) iff x+z≤y+w on the set ℝxℝ. It is reflexive because any real number can make x+z=y+w. It is not symmetric because if x+z≤y+w it's not possible for x+z≥y+w. It is...
  33. I

    Is the relation reflexive, symmetric, transitive

    Indicate which of the following relations on the given sets are reflexive on a given set, which are symmetric and which are transitive. {(x,y)\inZxZ: x+y=10} Tell me if I'm thinking about this correctly It is not reflexive because the only 5R5. It is symmetric because any xRy and yRx where...
  34. R

    Prove a rotationally symmetric central force field is conservative

    Homework Statement Suppose g:(0, +∞) → ℝ is continuous, and consider F:ℝd\{0} → ℝd, where F(x) = xg(|x|). Prove F is conservative. Homework Equations F is conservative iff there exists a C1 function f:ℝd\{0} → ℝd, s.t. F = grad(f). (edit: Or is the codomain of f actually ℝ, so that it's a...
  35. M

    Representation theory and totally symmetric ground state?

    Hello My question is about the ground state of vibrations for a solid. I'm working with graphite and have found out that for k=0 (The Gamma symmetry point), the vibrational modes can be decomposed into irreducible represenations in the following way Vibration = 2 * E1u + 2 * E2g + 2 * A2u...
  36. T

    Symmetry of the Universe: Why Doesn't 0 = cba?

    Why is it that our universe isn't perfectly symmetric? To demonstrate imagine that the 0 is the center of the universe: abc 0 cba Why doesn't the universe look like this? Why isn't there another me on the equally opposite side of the universe doing the same thing as I am right now?
  37. M

    Symmetric group S3 with symbols

    Homework Statement Determine the orders of all the elements for the symmetric group on 3 symbols S3. Homework Equations _______________________________________ The Attempt at a Solution 3 symbols : e,a,b I don't know how to do the S3 table using just these 3 letters I can do...
  38. Z

    Basic Symmetric Group Representation Question

    If you consider the permutation representation of Sn in ℂ^n, i.e the transformation which takes a permutation into the operator which uses it to permute the coordinates of a vector, then of course the subspace such that every coordinate of the vector is the same is invariant under the...
  39. T

    Solve Symmetric Group Homework: Find Subgroups of S6, S4 & S3 x S3

    Homework Statement Let G=S_6 acting in the natural way on the set X = \{1,2,3,4,5,6\}. (a)(i) By fixing 2 points in X, or otherwise, identify a copy of S_4 inside G. (ii) Using the fact that S_4 contains a subgroup of order 8, find a subgroup of order 16 in G. (b) Find a copy of S_3...
  40. S

    Symmetric, irreducible, tridiagonal matrix: Eigenvalues

    Homework Statement A) Let A be a symmetric, irreducible, tridiagonal matrix. Show that A cannot have a multiple eigenvalue. B) Let A be an upper Hessenberg matrix with all its subdiagonal elements non-zero. Assume A has a multiple eigenvalue. Show that there can only be one eigenvector...
  41. F

    What is the Proportion of Symmetric Matrices that have Positive Determinant?

    Homework Statement What proportion of 2x2 symmetric matrices with entries belonging to [0, 1] have a positive determinant? Homework Equations A^{T} = A If A = [[a, b], [c, d]] Then det(A) = ad - bc. But A is symmetric, so c = b. So det(A) = ad - b^2 So, in order for A to have a...
  42. M

    What Determines the Normalizer of a Sylow p-Subgroup in Sym(p)?

    Homework Statement What is the normalizer of the Sylow p-subgroup in the symmetric group Sym(p) generated by the element (1,2,...,p) where p is a prime number? Thanks Homework Equations na The Attempt at a Solution I know that the normalizer has order p(p-1). And I know that it has...
  43. Y

    MHB A and B are two symmetric matrices

    A and B are two symmetric matrices that satisfy: AB = - BA Which one of these statements are always true: a. (A-B)^2 is symmetric b. AB^2 is symmetric c. AB is invertable I tried to think of an example for such matrices, but couldn't even find 1...there must be a logical way to solve it...
  44. Demon117

    How to Show the Integral of a Spherically Symmetric Potential?

    Homework Statement Show that for a spherically symmetric potential \int _{all space} V(\vec{r})exp(i\vec{k}\cdot\vec{r})d\tau = \frac{4\pi}{r}\int_{0}^{\infty} V(r) sin(\kappa r)dr The Attempt at a Solution Given that the potential is spherically symmetric we have azimuthal symmetry...
  45. A

    SU(2)L, SU(2)R, other symmetric groups and SSB

    Hello everyone, When we speak about the SU(2)L group (in electroweak interactions for example), about what group do we talk ? What is the difference with the SU(2) group ? And with the SU(2)R ? Why is the label so important ? I ask this because I see that a Lagrangien can be invariant...
  46. G

    Linear Algebra Symmetric Matrix Set Question

    First of all, I apologize if this is in the wrong place. I didn't really know where it should be placed and if it is in the wrong place I am sorry. This question was on my recent Linear Algebra I final exam and I had no idea how to do it when I was writing the exam and I'm still stumped by...
  47. A

    Symmetric of a point relative to a line

    Homework Statement What is the easiest way of finding the symmetrical of a point relative to an arbitrary line? (I was asked on an exam to find the symmetrical of a point relative to the line y = x, but that's rather trivial - just switch the coordinates. How can I do it for any arbitrary...
  48. C

    Intro to Algebra, Symmetric group # of elements of order 4 in S6

    Homework Statement How many elements of order 4 are in S6? (symmetric group with order 6) Homework Equations The Attempt at a Solution So, the different forms of elements with order 4 in S6 are (abcd)(ef), (abcd) from there I am sunk on how to calculate. I know there are...
  49. N

    If the graph of a differentiable function is symmetric

    Homework Statement If the graph of a differentiable function f is symmertic about the line x=a, what can you say about the symmetry of the graph f'? Homework Equations The Attempt at a Solution
  50. A

    Is the Sign of a Permutation Zero if and Only if It's Not a Bijection?

    Homework Statement sgn(\sigma)=0\iff\sigma is not a bijection. The Attempt at a Solution (\rightarrow)Let sgn(\sigma)=0. Then, \Pi_{1\leq i<j\leq n}\frac{\sigma(j)-\sigma(i)}{j-i}=0. . For some i and j, i\neq j , \sigma(i)=\sigma(j) . Thus, \sigma is not an injection. (\leftarrow) \sigma...
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