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. G

    What is the Tensorial Property of Symmetry for Covariant Second Rank Tensors?

    How can I explain that the fact that a covariant second rank tensor is symmetric in one coordinate system is a tensorial property. This is for my GR course, but I didn't do a Tensor Calculus before.
  2. C

    Eigenvectors of symmetric matrices

    Can anyone prove that the eigenvectors of symmetric matrices are orthogonal?
  3. S

    On the definition of symmetric matrices

    Can a symmetric matrix contain complex elements(terms). If no, how is it that 'eigen values of a symmetric matrix are always real'(from a theorem) Is a symmetric matrix containing complex terms called a hermitian matrix or is there any difference? Can we call the following matrix...
  4. G

    Symmetric Difference Explanation

    Can someone explain to me how to show (x\y) union (y\x) = (x union y) \ (y union x) using only the main set theory laws for union, intersections and difference.
  5. M

    Symmetric Matrix and Definiteness

    Homework Statement If A is a symmetric matrix, what can you say about the definiteness of A^2? Explain. Homework Equations I believe I need to use the face that A^2=SD^2S^-1. I know that if all the eigenvalues of a symmetric matrix are positive then the matrix is positive...
  6. M

    Proving Symmetric Tensor Equation: S=[0.5(T+TT)]

    Homework Statement S : T = S:[0.5(T+TT)] S is a symetric tensor show for any tensor T the above is valid Homework Equations The Attempt at a Solution what i think i know ST=S S:T=tr[STT] =tr[ST] einstein notation tr[SijTjk] [SijTjk]ii but i can't really see this leading to...
  7. C

    Coding theory - binary symmetric channel

    Hi Let us suppose we transmit the binary digit '1'. The probability of not receiving '1' is p. Thus the probability of receiving '1' is 1-p. Suppose we send a longer code of length n. The probability of this code being received correctly is (1-p)^n. Now I don't understand this next...
  8. S

    Integral of an odd function over a symmetric interval

    Homework Statement f: [-a,a] >. R is Riemann integrable, prove that ∫[-a, a] ƒ (x) dx = 0 Homework Equations The Attempt at a Solution This only proof below I can think of is rather very calculus-ish.I wonder is there any other proof that is more Real Analysis level for this problem? Thanks...
  9. M

    What is the Q value for the symmetric fission of 236U?

    Homework Statement what is the Q value for the symmetric fission of 236U? Homework Equations M(Z,A)=Zmp+Nmn-B The Attempt at a Solution I don't understand the question by saying symmetric fission, is it mean we have the reaction which is 236U=118Ru+118Ru so the Q from he...
  10. M

    Proving Non-Cyclic and Non-Abelian Properties of Dihedral and Symmetric Groups?

    Homework Statement Prove that S_n and D_n for n>=3 are non-cyclic and non-abelian. Homework Equations I get that I need to show that two elements from each group do not commute and that there is not a single generator to produce the groups... I am just unsure of how to do this...
  11. M

    Center of Symmetric Groups n>= 3 is trivial

    Homework Statement The question is to show that the for symmetric groups, Sn with n>=3, the only permutation that is commutative is the identity permutation. Homework Equations I didn't know if it was necessary but this equates to saying the center is the trivial group. The Attempt at...
  12. Phrak

    Is General Relativity Time Symmetric?

    Is General Relativity Time Symmetric?
  13. M

    Ricci tensor: symmetric or not?

    I am really confused and the question can appear to be trivial or stupid: Is the Ricci tensor symmetric or anti-symmetric in a torsion-free affine connection? I am full of troubles since two different references gives two different answers (sorry no one is in english language but one of...
  14. F

    Reflexive and Symmetric Relations

    Homework Statement The set A has 5 elements. 1. How many relations exist on A? 2. How many of those relations are symmetric and reflexive? The Attempt at a Solution Some of the parts of this question are harder than others. 1. By simple counting, there are 2^(5^2) or 2^25 total relations...
  15. I

    Semi-Positive Definiteness of Product of Symmetric Matrices

    Here is my problem. Any ideas are appreciated. Let P be a projection matrix (symmetric, idempotent, positive semidefinite with 0 or 1 eigenvalues). For example, P = X*inv(X'*X)*X' where X is a regressor matrix in a least square problem. Let A be a symmetric real matrix with only integer...
  16. J

    How to find symmetric equations for the line of intersection of two planes?

    Hi, I have been at this single problem for two hours with nothing to show for it. Find symmetric equations for the line of intersection of the planes. z = 3x - y - 7 z = 4x + 2y - 6 They also give me one of the symmetric equations, z/10. I have over 3 pages of work for this. I...
  17. A

    A symmetric, transitive relation on a set that is not reflexive

    Can someone give an example of one? I can't think of one...
  18. P

    Symmetric difference problem (Real Analysis)

    Homework Statement What am I asked to do in the problem? Am I just asked to draw a diagram or to prove a) and b)? Homework Equations The Attempt at a Solution
  19. Rasalhague

    Metric Tensor & Symmetric Tensor Product in GR

    The Wikipedia article Metric tensor (general relativity) has the following equation for the metric tensor in an arbitrary chart, g = g_{\mu\nu} \, \mathrm{d}x^\mu \otimes \mathrm{d}x^\nu It then says, "If we define the symmetric tensor product by juxtaposition, we can write the metric in...
  20. D

    Understanding the Body Frame of a Spinning Symmetric Top

    Hello, I have a question about a spinning symmetric top: When the equations of motion are solved, they are solved in two frames--the space frame and the body frame. I understand the space frame, but in the body frame you are looking at the top from a frame that is rotating with it, right? So...
  21. M

    Spherically symmetric metric form

    spherically symmetric metric used to write in he following form: ds^2 = -h(r,t}^2 * dt^2 + f(r,t)^2 * dr^2 + r^2 * d_omega^2 But what about the form ds^2 = -f(r,t}^2 * dt^2 + f(r,t)^(-1) * dr^2 + r^2 * d_omega^2 and ds^2 = -f(r}^2 * dt^2 + f(r)^(-1) * dr^2 + r^2 * d_omega^2 how...
  22. S

    Confusion on anti-symmetric and symmetric

    confusion on "anti-symmetric" and "symmetric" Hi guys, I am a physics sophomore at next term, recently I am doing a reading on Naive Set Theory on my own. However, I got a few confusion. The books said that if A is a subset of B and B is a subset of A, then A=B, but this set inclusion is...
  23. D

    Lineal Algebra: Inverse Matrix of Symmetric Matrix

    Homework Statement Hello, I need some help in the fist parts of two lineal algebra problems, specially with algebraic manipulation. I guess that if I rewrite the determinant nicely some terms get canceled and I can write the inverse nicely, but don't know how to do it... Problem 1...
  24. Angelos K

    Definition of a symmetric connection

    Hi, all, According to my script, a connection \nabla_v is symmetric if the following holds (I assume for every pair of vectors): \nabla_v w - \nabla_w v =[v,w] What is the idea behind that? Why are we interested in that kind of symmetry (not for instance 0 instead of the commutator)...
  25. C

    Is a symmetric Lagrangian leads to a symmetric Stress-Energy Momentum？

    Is a symmetric Lagrangian leads to a symmetric Stress-Energy Momentum ？
  26. P

    Real Symmetric Endomorphism: Diagonalizability and Eigenvalues Explained

    Hi, We know that if u is a real symetric endomorphism, then u has a real eigenvalue and that u is diagonalizable. But can we say that u is diagonalizable with only real eigenvalues?
  27. mnb96

    Orthogonal and symmetric matrices

    Hello, I guess this is a basic question. Let´s say that If I am given a matrix X it is possible to form a symmetric matrix by computing X+X^{T} . But how can I form a matrix which is both symmetric and orthogonal? That is: M=M^{T}=M^{-1}.
  28. E

    Symmetric difference of set identity

    Is there a shorter way to verify this identity, as you can see I haven't even finished it. I know you can use Ven diagrams and truth tables but I wanted to avoid them inorder to use a more general formal approach. picture is attached
  29. C

    What is the difference between symmetric and antisymmetric relations?

    okay so i have looked up things online and they when other ppl explain it it still doesn't make sense. I am working on a few specific problems. R = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} the book says this is antisysmetric by sayingthat this relation has no pair of elements a and b with a...
  30. J

    How to Calculate Symmetric Relations in Set Theory?

    Hi. Let A = 1,2,3,4,5,6,7 How many symmetric relations on A contain exactly (a) four ordered pairs, (b) 5 , (c) seven and (d) eight The book has solutions to the first two, which I didn't understand at all. Please look the pic below Can someone guide me through how to approach the problem...
  31. M

    Show that for a symmetric or normal matrix

    Is there anyway to show that for a symmetric or normal matrix A, that det(A) = \prod \lambda_i without using Jordan blocks? I want to show this result using maybe unitary equivalence and other similar matrices... any ideas? It's obviously easy with JCF...
  32. K

    Conjugates in symmetric groups

    Homework Statement The question is, "How many conjugates does (1,2,3,4) have in S7? Another similar one -- how many does (1,2,3) have in S5? The Attempt at a Solution I know that the conjugates are all the elements with the same cycle structure, so for (123) I found there are 20...
  33. I

    Is the gcd function symmetric?

    Just a quick theory question. I'd assume it is, but usually the bigger number goes first. e.g. gcd(10, 5) = 2 but does gcd (5, 10) = 2? My guess is yes. Thanks for the help.
  34. B

    Prove Symmetric Matrixes Thm: A=0 or Skew Symmetric

    I need to prove the following. 1. A is a symmetric matrix, and x(transpose)*A*x=0 for all x (belongs to R^n) if and only if A=0. 2. x(transpose)*A*x=0 for all x (belongs to R^n), if and only if A is skew symmetric.
  35. E

    Metric outside a spherically symmetric source.

    I've been learning GR in my sparetime, and occationally I run into a conceptual problem that stalls my progress. Here is a question that has come up. I expect that this is a stupid question, but it's really bugging me, and an explanation will help me move forward more efficiently. If we wish...
  36. J

    Why is the ground state always symmetric?

    Homework Statement Why is the ground state always symmetric and first excited state anti-symmetric? OR Why does the ground state always have no node and first excited state has one node? Homework Equations The Attempt at a Solution
  37. Phrak

    Can a Symmetric Tensor on a Manifold of Signature -+++ be Written in p-forms?

    Electric charge continuity is expressed as ∂tρ + ∂iJi =0. (1) The manifold, M in question is 3 dimensional and t is a parameter, time. ∂iJi is the inner product of the ∂ operator and J. With M a subspace of a 4 dimensional manifold with metric signature -+++, eq. (1) can be written in...
  38. F

    Question about double (and triple) integrals over a symmetric area

    ]This isn't a home work question in particular, but just want confirmation about a general idea. So in Calc III, you have integrals of the form \int_{-a}^a \int_{-\sqrt{a^2 - x^2}}^{\sqrt{a^2 - x^2}} x y dy dxwhich is the typical rectangular coordinates for a circle. Now, the integrand is the...
  39. I

    Symmetric relation on ordered pairs

    Homework Statement For sets A and B, define a relation \mathcal{R} on A∪B by: \forall A, B \in A \cup B, x\mathcal{R}y if and only if (x,y) \in A \times B For all sets A and B, if R is symmetric, then A = BHomework Equations The Attempt at a Solution I tried doing this, and I heard it's...
  40. M

    Spherically symmetric potential and spherical harmonics

    When solving the time-independent Schrodinger equation for a spherically symmetric potential, using the separation of variables, we find that solutions of the form \psi =R(r)Y_l^m(\theta ,\phi) where the Y_l^m are the spherical harmonics. We apply this to the (idealized) electron in a Hydrogen...
  41. M

    Maple Symmetric polynomials in Maple?

    Does anyone know if it possible to generate elementary symmetric polynomials in Maple (I am using version 12), and if so, how? I have scoured all the help files, and indeed the whole internet, but the only thing I have found is a reference to a command "symmpoly", which was apparently...
  42. N

    Condition on Leading principle minors of a symmetric Positive semidefinite(PSD) matri

    Hi everyone, This is related to my previous https://www.physicsforums.com/showthread.php?t=392069" Let A=(a_{ij}) be a symmetric (i.e., over reals) PSD matrix with the following conditions on Leading Principle Minors (determinant of the submatrix consisting of first i rows and i...
  43. N

    A condition on principle minors of a symmetric Positive semidefinite (PSD) matrix

    Hi everyone, Let A=(a_{ij}) be a symmetric (i.e., over reals) PSD matrix. Then is the following correct? "If any principle minor ( \ne A ) be zero, then all principle minor contained in this minor should also be zero". I can not prove or disprove it..any help? By the way how...
  44. M

    Symmetric polynomial algorithm?

    Let f, g \in \mathbb{Z}[x, y, z] be given as follows: f = x^8 + y^8 + z^6 and g = x^3 +y^3 + z^3. Express if possible f and g as a polynomial in elementary symmetric polynomials in x, y, z. Professor claims there is an algorithm we were supposed to know for this question on the midterm. I...
  45. N

    Mass conservation in radially symmetric parabolic PDE problems

    Dear all, I'm trying to solve the 2d heat equation in a radially symmetric domain, numerically using the Crank-Nicolson method. i.e. \dfrac{\partial u}{\partial t} = D\left( \dfrac{\partial^2u}{\partial r^2}+\dfrac{1}{r}\dfrac{\partial u}{\partial r}\right) Applying the Crank-Nicolson...
  46. G

    Explaining Symmetric Input in Instrumentational & Differential Amplifiers

    I would like to ask you what it is meant by "symmetric input" in the instrumentational amplifier schematics and in the differential amplifier? I can`t understand what is the difference between symmetric and non symmetric input and output as parameters. Can anyone explain ? Thanks
  47. E

    Show the Symmetric group is generated by the set of transpositions (12) (n-1 n).

    Homework Statement 5.1: Prove that S_n is generated by the set {(1 2), (3 4),...,(n-1 n)}Homework Equations None that I know ofThe Attempt at a Solution Any element in S_n can be written as a product of disjoint n-cycles. So now I need to show any n-cycle can be written as a product of...
  48. T

    Understanding the Symmetry Property of Relations in Velleman's 'How to Prove It

    Homework Statement In Velleman's "How to Prove it", he gives a proof that "R is symmetric iff R = R-1, which I find to be confusing when he is proving that R^{-1}\subseteq{R}: Now suppose (x,y)\in R^{-1}. Then (y,x)\in R, so since R is symmetric, (x,y)\in R. Thus, R^{-1}\subseteq R so R=R-1...
  49. K

    Determinant of a symmetric matrix

    Hi, Is there a simplification for the determinant of a symmetric matrix? For example, I need to find the roots of \det [A(x)] where A(x) = \[ \left( \begin{array}{ccc} f(x) & a_{12}(x) & a_{13}(x) \\ a_{12}(x) & f(x) & a_{23}(x) \\ a_{13}(x) & a_{23}(x) & f(x) \end{array}...
  50. I

    Proving Identity for Non-Zero Symmetric Covariant Tensors

    Homework Statement For ease of writing, a covariant tensor \bf G.. will be written as \bf G and a,b,c,d are vectors. Let \bf S and \bf G be two non-zero symmetric covariant tensors in a four-dimensional vector space. Furthermore, let S and G satisfy the identity: [\bf G \otimes \bf...
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