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

    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...
  2. 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}...
  3. 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...
  4. K

    Exploring Odds and Probability of a Symmetric Coin Toss

    Hi everyone There is a question which I find very hard to solve and it goes like this.. A symmetric coin with heads on one side and tails on the other side is tossed 491 times after one another. The total amount of times you get tails is either even or odd. Is the probability that you get...
  5. C

    Dot product of vector and symmetric linear map?

    Homework Statement My book states as follows: --- If u and v have the coordinate vectors X and Y respectively in a given orthonormal basis, and the symmetric, linear map \Gamma has the matrix A in the same basis, then \Gamma(u) and \Gamma(v) have the coordinates AX and AY, respectively. This...
  6. J

    Why is Cramer's rule for determinants not 'symmetric'?

    we can solve non-homogeneous equations in matrix form using Cramer's rule. This rule is valid only if we are replacing the columns. Why can't we replace the rows and carry on the same? For eg we can use elementary transformations for obtaining inverses either via rows or via columns. But we...
  7. S

    Find parametric equations and symmetric equations for the line

    Homework Statement Find parametric equations and symmetric equations for the line through P0 and perpendicular to both given vectors. (P0 corresponds to t = 0.) P0 = (1, 1, 0) i + j and j + k Homework Equations The Attempt at a Solution For the symmetric equations, I did...
  8. M

    Fermions must be described by antisymmetric and bosons by symmetric

    :confused: friends, we know that fermions must be described by antisymmetric and bosons by symmetric wavefunctions. but i was wondering why a particle of certain class behaves like that for ever? ie. say, an electron will never behave like a boson ?? my book says that there is a spin...
  9. P

    Scalar fields: why symmetric ener-mom. tensor?

    I'm studying the properties of the energy momentum tensor for a scalar field (linked to the electromagnetic field and corresponding energy-momentum tensor) and now I'm facing the statement: "for a theory involving only scalar fields, the energy-momentum tensor is always symmetric". But I've...
  10. T

    Proving Real Eigenvalues for Symmetric Matrix Multiplication?

    Homework Statement Given a real diagonal matrix D, and a real symmetric matrix A, Homework Equations Let C=D*A. The Attempt at a Solution How to prove all the eigenvalues of matrix C are real numbers?
  11. T

    Help Prove Real Eigenvalues of Symmetric Matrix

    Help! Symmetric matrix I know that all the eigenvalues of a real symmetric matrix are real numbers. Now can anyone help out how to prove that "all the eigenvalues of a row-normalized real symmetric matrix are real numbers"? Thank you~~~
  12. M

    Determine if the following problem is symmetric and transitive

    Homework Statement Suppose ~ is defined on the whole numbers by a~b iff ab2 is a perfect cube. Determine if ~ is symmetric transitive Homework Equations ab2 must ba2 The Attempt at a Solution I tried using different numbers, but it isn't coming out as a perfect square. For...
  13. A

    Axially symmetric B field vector potential?

    Suppose you have an axially symmetric magnetic field for which the azimuthal component B_\phi = 0. This is all you know. What are some possible vector potentials \vec A (such that \vec B = \nabla \times \vec A) that would produce this field? (So we're working in cylindrical coordinates.) The...
  14. A

    What does axially symmetric mean mathematically?

    What does "axially symmetric" mean mathematically? If we, for example, say that a magnetic field \vec B is axially symmetric, does that mean that (in cylindrical coordinates) we have \frac{\partial \vec B}{\partial \phi} = 0, where \phi is the azimuthal angle?
  15. C

    Confirming Symmetric & Antisymmetric Solutions for Wave Function

    Homework Statement Hello, Can you confirm that what I wrote is correct for the given potential? https://www.physicsforums.com/attachment.php?attachmentid=22309&stc=1&d=1260118852 Now I wrote the term for the wave funcation and for the given symetric potential , the functions of the...
  16. P

    Symmetric matrix real eigenvalues

    Homework Statement Prove a symmetric (2x2) matrix always has real eigenvalues. The problem shows the matrix as {(a,b),(b,d)}. Homework Equations The problem says to use the quadratic formula. The Attempt at a Solution From the determinant I get (a-l)(d-l) - b^2 = 0 which...
  17. H

    Do 3x3 Symmetric Matrices Form a Subspace of 3x3 Matrices?

    symmetric matrices... help please! hi can someone tell me...how to correctly use the 10 axioms.. for example: does the set of all 3x3 symmetric matrices form a subspace of a 3x3 matrices under the normal operations of matrix addition and multiplication? I don't really get how to prove this..
  18. T

    Proving Sum of Symmetric & Skew-Symmetric Matrix

    Homework Statement Prove that any square matrix can be written as the sum of a symmetric and a skew-symmetric matrix Homework Equations For symmetric A=A^{T} For scew-symmetric A=-A^{T} The Attempt at a Solution Not sure where to begin. Using algebra didn't work. Got powers...
  19. S

    Prove A is symmetric iff x*Ay = Ax*y

    Hello everyone. This is my first official post here but I have been lurking around for about a year now. Homework Statement Prove that a matrix A is symmetric if and only if x*Ay = Ax*y for all x,y of R^n, where * denotes the dot product. Homework Equations The Attempt at a...
  20. A

    Bridge rectifier and symmetric power supply

    Could some one explain how a bridge rectifier works with a diagram and its mathematics? Also please explain how a center tap transformer and a bridge rectifier are used to provide a symmetric power supply. I have the Ckt Diagram for that but I cannot understand its working. thanks...
  21. B

    Isomorphism between Dihedral and Symmetric groups of the same order?

    Is there a way to prove generally that the Dihedral group and its corresponding Symmetric group of the same order are isormorphic. In class we were only shown a particular example, D3 (or D6 whatever you wish to use) and S3, and a contructed homomorphism, but how could you do it generally? Would...
  22. N

    Proof regarding skew symmetric matrices

    Homework Statement Show that if A is skew symmetric, then Ak is skew symmetric for any positive odd integer k. Homework Equations The Attempt at a Solution Wow, I have no idea how to prove this. I'm guessing there's going to be induction involved. I know that the base case of k...
  23. Born2bwire

    Solving Gen. Eigen Probs w/ Real Sym Indefinite A & Definite B

    I have a system that ideally creates a real symmetric negative definite matrix. However, due to the implementation of the algorithm and/or finite-precision of floating point, the matrix comes out indefinite. For example, in a 2700 square matrix, four eigenvalues are positive, the rest are...
  24. J

    Skew Symmetric Determinant Proof

    Hi all! I was working on some homework for the linear algebra section of my "Math Methods for Physicists" class and was studying skew symmetric matrices. There was a proof I saw on Wikipedia that proves that the determinant of a skew symmetric matrix is zero if the number of rows is an odd...
  25. M

    What are half wave symmetric waves ?

    what are half wave symmetric waves ? hello friends...i m studying signal n system...i havnt find suitable info about half wave symmetric waves from anywhere...i need to understand that how to find whether any wave is half wave symmetric or not...please help me friends ...
  26. C

    Basis for set of 2x2 complex symmetric matrices

    Homework Statement Give the basis and dimension of the set of all 2x2 complex symmetric matrices. Homework Equations The Attempt at a Solution I know that if the coefficients were real, then I could just have the basis \left( \begin{array}{cc} 1 & 0\\ 0 & 0 \end{array}...
  27. M

    Levi-civita and symmetric tensor

    Homework Statement Show that \epsilon_{ijk}a_{ij} = 0 for all k if and only if a_{ij} is symmetric.Homework Equations The Attempt at a Solution The first bit I think is just like the proof that a symmetric tensor multiplied by an antisymmetric tensor is always equal to zero. \epsilon_{ijk} = -...
  28. L

    How Do You Solve the Symmetric Delta Potential Problem in Quantum Mechanics?

    This problem is a symmetric delta potential problem that I was given a few days ago and I can't seem to get the gist of it. Question: Find the spectrum and wave functions of a particle in the potential V(x)=G[d(x-a)+d(x-a)] Calculate the transmission and reflection amplitude. Where G can be...
  29. M

    Exterior calculus: what about symmetric tensors?

    Hi all, Quick question I haven't been able to find the answer to anywhere: Can I use exterior calculus for symmetric tensors? I'm familiar with the exterior calculus approach to things like Stokes's theorem and Gauss's law, but that's vector stuff. It seems to me the only tensors in...
  30. V

    3D wave equation - spherically symmetric transformations

    Problem: Applied Partial Differential Equations (Richard Heberman) 4ed. #12.3.6 Consider the three dimensional wave equation \partial^{2}u/\partial t^2 = c^2\nabla^2 u Assume the solution is spherically symetric, so that \nabla^2 u =...
  31. S

    Question about spherically symmetric charged objects

    Hi, I would like to ask a question about spherically symmetric charged objects. My teacher told me that you can treat spherically symmetric charged objects at point charges. However, my teacher did not prove it. I guess you have to integrate every small volume on the spherically symmetric...
  32. W

    How Do Symmetric Graphs Relate to Equations and Calculations?

    1. See Attachments 2. None 3. 1st Attachment #19 I believe that I am suppose to multiply (x-2)(x+2) but what do i do about the symmetric with an origin? 2nd Attachment I do not get what they are asking for in the 2nd and 3rd part of the question can you please explain it to me...
  33. N

    A particle in a spherical symmetric potential

    Homework Statement A particle that moves in three dimensions is trapped in a deep spherically symmetric potential V(r): V(r)=0 at r<r_0 infinity 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)...
  34. M

    Consider the heat equation in a radially symmetric sphere of radius

    Consider the heat equation in a radially symmetric sphere of radius unity: u_t = u_{rr}+{2 \over r}u_r \ for \ (r,t) \in (0,1) x (0,\infty) with boundary conditions \lim_{r \rightarrow 0}u(r,t) < \infty ; \ u(1,t)=0\ for \ t >0 Now, using separation of variables u=R(r)T(t) leads to the...
  35. H

    Is a symmetric matrice allways nonsingular?

    Can someone confirm this? If so, are there any respected websites on the net that can confirm this theorem?
  36. C

    Is a Matrix Symmetric if Row Space Equals Column Space?

    I had this question on a test and I was wondering why it is false: If the row space equals teh column space then AT=A.
  37. A

    Explain what this tells you about parametric and symmetric equations in R^3?

    parametric and symmetric equations in R^3?? Homework Statement Recall that there are three coordinates planes in 3-space. A line in R3 is parallel to xy-plane, but not to any of the axes. Explain what this tells you about parametric and symmetric equations in R3. Support your answer using...
  38. malawi_glenn

    Eigenvalues and eigenvectors of symmetric 2x2 matrix?

    Hello I recall, I think, that there is a lemma which states that a 2x2 symmetric matrix can be diagonalized so that its eigenvalues are (trace) and 0. I can not find it anywhere =/ I think it was a physics teacher who told us this a couple of years ago, can anyone enlighten me? cheers
  39. P

    Exponential of a tridiagonal symmetric matrix

    Guys does anyone know of a technique to find the exponential of a tridiagonal symmetric matrix... Thanks in advance
  40. P

    Is Flatness a Self-Selected Outcome in a Maximally Symmetric Universe?

    when a space (or spacetime) is said to be maximally symmetric, does this mean that it is homogeneous?
  41. O

    Sylow Subgroups of Symmetric Groups

    Homework Statement Find a set of generators for a p-Sylow subgroup K of Sp2 . Find the order of K and determine whether it is normal in Sp2 and if it is abelian. Homework Equations The Attempt at a Solution So far I have that the order of Sp2 is p2!. So p2 is the highest power of...
  42. Amith2006

    Cylindrically symmetric potential function

    Homework Statement I am trying solve the 3-D Schrodinger equation for a particle in a cylindrically symmetric potential. If it was the case of spherically symmetric potential, then we can approximate it to a central potential. But what will be the form of the potential in the cylindrically...
  43. 3

    "Symmetric Relations: Is "Is Brother of" Symmetric?

    Homework Statement Is this relation symmetric? The relation in a set of people, "is brother of" Homework Equations aRb , bRa The Attempt at a Solution The answer is not symmetric. They gave example says that paul may be the bother of Anne but Anne is not the brother of paul...
  44. H

    Twin Grain Boundary and Symmetric Tilt Grain boundary.

    Is there a difference between Twin grain boundary and symmetric tilt grain boundary? If so, what is it?
  45. Y

    Symmetric matrix with eigenvalues

    Homework Statement Let {u1, u2,...,un} be an orthonormal basis for Rn and let A be a linear combination of the rank 1 matrices u1u1T, u2u2T,...,ununT. If A = c1u1u1T + c2u2u2T + ... + cnununT show that A is a symmetric matrix with eigenvalues c1, c2,..., cn and that ui is an eigenvector...
  46. T

    A presentation for the symmetric group

    Homework Statement How to find such a presentation? Where would one start?
  47. A

    Understanding the Symmetric Matrix Problem: A Brief Overview

    Homework Statement consider the 2*2 symmetric matrix A = (a b ) (b c) and define f: R^2--R by f(x)=X*AX . show that \nablaf(x)=2AX Homework Equations The Attempt at a Solution quiet confuse about this question \nablaf(x)=(Homework Statement consider the 2*2 symmetric matrix A...
  48. N

    Optimum (min/max) of a symmetric function

    Hi all, I'm wondering if the following argument is right: "The optimum (minimum/maximum) value of a symmetric function f(x_1,x_2,...,x_n) (By 'symmetric' I mean that f remains same if we alter any x_i's with x_j's), if exists, should be at the point x_1=x_2=...=x_n". Please help me by...
  49. K

    Dimension of a set of symmetric matrices & prove it's a vector space

    Prove: the set of 3x3 symmetric matrices is a vector space and find its dimension. Well in class my prof has done this question, but I still don't quite get it.. Ok, first off, I need to prove that it's a vector space. The easy way is probably to prove that it contains the zero space and...
  50. N

    Proving Maximal Symmetric Extension of a Symmetric Operator

    A symmetric operator has a maximal symmetric extension. How do you prove this?
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