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. A

    Real Symmetric Positive Definite Matrices

    Homework Statement Let A be a real symmetric positive definite matrix. Show that |aij|<(aii+ajj)/2 for i not equal to j. Homework Equations The Attempt at a Solution I really don't even know where to start with this. I think that aii and ajj must both be > 0 since they are on the...
  2. D

    On the invertibility of symmetric Toeplitz matrices

    I am curious if anyone knows conditions on the invertibility of a symmetric Toeplitz matrix. In my research, I have a symmetric Toeplitz matrix with entries coming from the binomial coefficients. Any help would be appreciated. Ex: [6 4 1 0 0] [4 6 4 1 0] [1 4 6 4 1] [0 1 4 6 4] [0...
  3. L

    Symmetric Matrices to Jordan Blocks

    I've been working through the Linear Algebra course at MITOCW. Strang doesn't go into the Jordan form much. When a matrix A is diagonalizable then A= S \Lambda S^{-1} and the matrix S can be formed from eigenvectors that correspond to the eigenvalues in \Lambda Question: how do...
  4. G

    Is there a way to diagonalise a tridiagonal symmetric matrix?

    The matrix A is symmetric and tridiagonal. If B is the matrix formed from A by deleting the first two rows and columns, show that \left|A\right| = a_{}11\left|M_{}11\right| - (a_{}1)^{}2\left|B\right| where \left|M_{}11\right| is the minor of a_{}11 I know what a symmetric tridiagonal...
  5. X

    Why does the system has lower energy if its wave function is symmetric?

    Hi all: I am confused that in general case, if [H,p]=0 (where H is Hamiltonian of system and P is parity operator), system wave function is either symmetric or antisymmetric. How do we know that system is in lower energy state if its wave function is symmetric by comparing that system is...
  6. M

    Perturbation Theory with Symmetric Rotator

    Homework Statement Given the Hamiltonian and perturbation below, what are the energy shifts for the states with l=1 Given H_{0}=(L^2)/(2I) H_{1}=E_{1}cos\vartheta Homework Equations L= r x P The Attempt at a Solution in order to find the first order correction to the energy...
  7. S

    What Metrics Can Be Defined on a Symmetric Group Beyond the Discrete Metric?

    If I convert a symmetric group of degree n into a metric space, what metrics can be defined except a discrete metric? If a metric can be defined, I am wondering if the metric can describe some characteristics of a symmetric group.
  8. D

    Understanding Symmetric Matrix Properties: A Puzzling Example

    Homework Statement http://img266.imageshack.us/img266/152/78148531ur5.png Homework Equations A is symmetric. The Attempt at a Solution First of all if you calculate rT you'll get qTA so why it the order reversed in the picture above? Moreover I don't see why it is zero.
  9. S

    Symmetric functions/odd even or neither

    Homework Statement I am supposed to find out if this function is symmetric with reference to the y-axis or reference to the origin. The function is F(x)=4x^2/(x^3+x) Homework Equations These are how to know if even or odd A(-x)^even power = ax^even power A(-x)^odd power = -ax^odd power The...
  10. B

    Solving a Puzzling Problem: No A Exists for Symmetric Matrix B

    Give an example of a 2X2 symmetric matrix B that cannot be written as B = ATA. Give an explanation as to why no such A exists for the matrix B you have given. I know that the product ATA is a symmetric matrix, but how could there be no such A that exists for some matrix B? I'm really...
  11. S

    Symmetric group (direct product and decomposition)

    I am looking for a mechanism to find a decomposition of symmetric groups. For finitely generated abelian group G, there is a mechanism to decompose G such that G is isomorphic to a direct sum of cyclic groups. For symmetric groups, it seems a bit complex for me to find it. For example...
  12. S

    Symmetric Matrix as a subspace

    My question is; Let S = {A € Mn,n | A = AT } the set of all symmetric n × n matrices Show that S is a subspace of the vector space Mn,n I do not know how to start to this if you can give me a clue for starting, I appreciate.
  13. B

    Points that are symmetric with respect to a circle C

    Homework Statement Lemma 1: Fix the circle C with center (x nought, y nought); y nought is greater than 0 and radius R is less than y nought. Consider two points P (x nought, y noight tilde) and P prime (x nought, -y nought tilde) which are symmetric with respect to x-axis by construcion...
  14. P

    Is the Momentum-Energy Tensor Always Symmetric?

    I was reading about the momentum-energy tensor (or stress-energy tensor), at one point the author says, " \theta^{\mu\nu} = (\partial^\mu\phi)(\partial^\nu\phi) - g^{\mu\nu}L This is clearly symmetric in \mu and \nu." \theta^{\mu\nu}: is the stress-energy tensor \phi is a scalar field...
  15. S

    Euler angles and symmetric top

    Homework Statement Check out problem 5.7 part a I want to express the exterior gravitational potential in terms of the Euler angles so that I might eventually use (dV/dB)B=B0 = 0 - the condition for equilibrium. I am therefore expecting the Lagrangian to be cyclic in terms of the other two...
  16. G

    Solving Symmetric Tensor: c\cdot (A \times b) \neq (A \times b) \cdot c

    Homework Statement Demostrate: c\cdot (A \times b) \neq (A \times b) \cdot c with A \in\Re^{3 \times 3} is a symmetric Tensor of second order and b,c \in \Re^3 are vectors Homework Equations The Attempt at a Solution (A \times b)_ {ij} = A_{ij} \epsilon _{jkl} b_l
  17. K

    Proving Symmetry of AAT: Is it Possible?

    Homework Statement A matrix S is symmetric if S = ST. Show that AAT is symmetric for any matrix A. Homework Equations AAT = (AT)TAT The Attempt at a Solution I just said: A = (AT)T and AT = AT Therefore AAT is symmetric. I am unsure if that proves it or if I just went in...
  18. M

    Modelling a flow past a symmetric body

    Homework Statement I need to model the streamlines for an incompressible, irrotational flow about various symmetric bodies, starting with an ellipseHomework Equations for a uniform flow in the positive x direction, psi =y for a source/sink of strength K at (xn,yn) psi = Karctan((y-yn)/(x-xn))...
  19. F

    Analyzing a Cylindrically Symmetric Plasma Column

    Homework Statement A cylindrically symmetric plasma column in a uniform B field (= B0 in z direction) has n(r) = n0 exp[-(r/r0)^2] and ni = ne = n0exp[e phi/kb Te] where phi is the potential and Te is the temp of the electrons. (a) Show that Ve and VDe are equal and opposite Homework...
  20. L

    Properties of 4x4 symmetric matrix with eigvals E1, -E1, E2, -E2

    Hi there, I would appreciate if you could share your exeriences or ideas about properties of 4x4 symmetric/hermitean matrices H such that U^T H U = D = diag( E1, -E1, E2, -E2 ) or diag (E1, E2, -E1, -E2 ) The things I would like to perform are the following - decompose an expression...
  21. B

    A little help with symmetric, reflexive and transitive

    Hello, It's been a while since I've had to determine whether a relation is reflexive, symmetric or transitive so I would appreciate a bit of guidance. I now that being reflexive means the relation xRx for all x and being symmetric implies xRy and yRx for all x, y and transitive implies that if...
  22. F

    Prove the property of skew symmetric matrix

    Homework Statement Hi, I need to prove that if S is a skew-symmetric matrix with NXN dimension and B is any square real-valued matrix, therefore the product of transpose(B), S, and B is also askew symmetric matrix Homework Equations This is what I know so far. 1.Transpose(S) = -S...
  23. T

    Is A Skew Symmetric?

    Let A in n x n real matrix. For every x in R^n we have <Ax,x>=0 where < , > is scalar product. prove that A^t=-A (A is skew symmetric matrix)
  24. L

    Write parametric and symmetric equations for the z-axis.

    Write parametric and symmetric equations for the z-axis. I'm not sure i am on the right track; here is my attempt to an answer. [0, 0, z] where z can equal any number. a = [0, 0, 1] b = [0, 0, z] Parametric equations x = 0 y = 0 z = 1 + tz Symmetric equations...
  25. N

    Measure theory and the symmetric difference

    Hi, I'm currently trying to teach myself some measure theory and I'm stuck on trying to show the following: Let (X,M,\mu) be a finite positive measure space such that \mu({x})>0 \forall x \in X . Set d(A,B) = \mu(A \Delta B), A,B \in X. Prove that d(A,B) \leq d(A,C) + d(C,B) . Could...
  26. CalleighMay

    Symmetric equations of tangent lines to curves

    Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me? The problem is on pg 950 in chapter 13.7 in the text, number 46. It...
  27. N

    Understanding Symmetric Groups: S4 Order & Products

    What is the order of S4, the symmetric group on 4 elements? Compute these products in order of S4: [3124] o [3214], [4321] o [3124], [1432] o [1432]. Can I get help on how to do this. The solution's manual gives the answer on how to do the last two, but I don't understand the process...
  28. S

    A weird spherically symmetric metric

    a weird "spherically symmetric" metric Minkowski metric in spherical polar coordinates [t, r, theta, phi] is ds^2 = - dt^2 + dr^2 + r^2\,(d\theta^2 + sin^2(\theta)\, d\phi^2). The question is what happens when the coefficient of the angular part is set to constant, say 1, instead of r^2...
  29. N

    Reflexive, Symmetric, or Transitive

    Determine whether the following digraph represents a relation that is reflexive, symmetric, or transitive. Not sure how to determine this. Any help would be wonderful. The digraph is uploaded into a word document.
  30. Q

    Can the Symmetric Twin Paradox be Tested with Atomic Clocks?

    So, I was thinking about a variation on the Twin Paradox, and was hoping someone could help me work through it. The motivation is the usual explanation for the Twin Paradox, namely that one twin accelerates and so breaks the symmetry. This begs the question of what happens when both twins ride...
  31. E

    What is the Center of the Symmetric Group when n ≥ 3?

    [SOLVED] Center of Symmetric Group Homework Statement Show that for n ≥ 3, Z(Sn) = {e} where e is the identity element/permutation. The attempt at a solution It is obvious that e is in Z(Sn). If there is another element a ≠ e in Z(Sn), then... There must be some sort of contradiction and...
  32. T

    Self-Reproducing Rays in a symmetric Resonator

    Homework Statement Self-Reproducing Rays in a symmetric Resonator. Consider a symmetric resonator using two concave mirrors of radii R separated by a distance d=3|R|/2. After how many round trips through the resonator will a ray retrace its path? Homework Equations The Attempt at...
  33. MathematicalPhysicist

    Can a subset of R^n have multiple centres of symmetry?

    X a subset of R^n is called centrally symmetric if the isometry f_z:R^n->R^n defined by x|->2z-x for some z in R^n satisifies: f_z(X)=X. and z is called centre of symmetry. Now i need to show that: 1. if X is centrally symmetric and f is an isometry then f(X) is also centrally symmetric. 2...
  34. P

    What is the third condition for finding the symmetric point of a line?

    Homework Statement Find the coordinates of the symmetric point of the point M(2,1,3) of the line \frac{x+2}{1}=\frac{y+1}{2}=\frac{z-1}{-1} Homework Equations The Attempt at a Solution Out from here: \frac{x-x_1}{x_2-x_1}=\frac{y-y_1}{y_2-y_1}=\frac{z-z_1}{z_2-z_1}...
  35. P

    What is the symmetric point of the point M(3,4,7) from the plane 2x-y+z+9=0?

    Homework Statement Find the coordinates of the symmetric point of the point M(3,4,7) from the plane 2x-y+z+9=0 Homework EquationsThe Attempt at a Solution I found the equation of the plane which the symmetric point is staying at: 2x-y-z+27=0 Also I found the distance between M(3,4,7) and...
  36. S

    Matrix relation of sets. symmetric, antisymmetric,reflexive,transitive

    Homework Statement relation A = {a,b,c} for the following matrix [1,0,0;1,1,0;0,1,1] is it reflexive, transitive, symmetric, antisymmetric Homework Equations ordered pairs. The Attempt at a Solution i wrote the ordered pairs as (a,a),(b,a),(b,b),(c,b),(c,c) I only that...
  37. I

    Examining Forces in a Symmetric Building

    Homework Statement A symmetric building has a roof sloping upward at 34.0 degrees above the horizontal on each side. A)If each side of the uniform roof weighs 1.10×10^4N , find the horizontal force that this roof exerts at the top of the wall, which tends to push out the walls...
  38. M

    Gauss's Law to Symmetric Charge Distribution

    A 10.0 gram piece of styrofoam carries a net charge of -0.700\muC and floats above the center of a large horizontal sheet of plastic that has a uniform charge density on its surface. What is the charge per unit area on the plastic sheet? Homework Equations \Phi = E\intdA =...
  39. O

    MATLAB, eigenvalues and condition number of a symmetric square matrix

    2. Write a MATLAB® function to calculate the condition number of a symmetric square matrix of any size by means of Eigenvalues: § The power method should be used to calculate the Eigenvalues. § The script (function) should give an error message if the matrix is not...
  40. N

    Proving Trigonometric E-Values for a Symmetric Tridiagonal Matrix

    Homework Statement A = [ b c ... 0000000000000000000 ] [ c b c ... .000000000000000 0] [ ... ] [ 000000000000000000 c b c ] [ 000000000000000000 b c ] where a,b are real. This matrix is tridigonal and symmetric. I need to...
  41. Peeter

    Doran/Lasenby. Commutator and symmetric products?

    Geometric Algebra for Physicists, in equation (4.56) introduces the following notation A * B = \langle AB \rangle as well as (4.57) the commutator product: A \times B = \frac{1}{2}\left(AB - BA\right) I can see the value defining the commutator product since this selects all...
  42. S

    Improving a Search Heuristic for symmetric number-board game

    Board (4x3) A B C D E F G H I J K L You have 11 chips [1,1,1,1], [2,2,2,2], [3,3], [4] (for example) placed randomly on the 4x3 board in slots A,...,L and one empty slot (that moves) that we'll call X, the movements are done similarly to the 8-puzzle game where X (empty) can move either...
  43. S

    Improving a Search Heuristic for symmetric number-board game

    Board (4x3) A B C D E F G H I J K L You have 11 chips [1,1,1,1], [2,2,2,2], [3,3], [4] (for example) placed randomly on the 4x3 board in slots A,...,L and one empty slot (that moves) that we'll call X, the movements are done similarly to the 8-puzzle game where X (empty) can move either...
  44. P

    Root of the symmetric equation

    Homework Statement Solve this equation, and find x. 6x^5-5x^4-29x^2-5x+6=0 Homework Equations if x= \alpha is root of the symmetric equation, then x= \frac{1}{\alpha}, is also root of the symmetric equation The Attempt at a Solution I tried first to write like this...
  45. M

    Symmetric object prove principle axis goes through CM

    Homework Statement c) For such a "symmetric" object, prove that any axis going through the center of mass is a principal axis. Homework Equations The Attempt at a Solution I am not sure how they want me to prove this. I was looking at the Displacement Axis Thm but I am not sure...
  46. tony873004

    Parametric and symmetric equations

    Find the parametric and symmetric equations of the line of intersection of the planes x+y+z=1 and x+z=0. I got the normal vectors, <1,1,1> and <1,0,1> and their cross product <1,0,-1> or i-k. I set z to 0 and got x=0, y=1, z=0. How do I form parametric equation out of this?? I know...
  47. G

    Solving electrostatic, rotationally symmetric 3D problem with conformal mapping?

    I heard that one can solve 2D problem with conformal mapping of complex numbers. Is it possible to use this method for 3D axial-rotationally symmetric problems (which are effectively 2D with a new term in the differential equation)?
  48. K

    Given a real nxn symmetric and non-positive definite matrix,. .

    let B be a nXn real symmetric and non-positive definite matrix. Show that (x^TBx)^1/2 is not a norm on R^n.
  49. M

    Symmetric Potentials - Eigenstates & Ground States

    Hi, Can anyone help me to understand the following please? If a potential is symmetric does this mean that the eigenstates are either symmetric or antisymmetric? Is the ground state always symmetric and the first excited state always antisymmetric? Thanks!
  50. A

    What does 'M symmetric' mean in the context of matrices?

    My dad came across this phrase in a book but neither of us are familiar with it. The statement is : "Let M_{1} and M_{2} be matrices. N = M_{1}^{-1}M_{2}. This matrix is M_{1} symmetric and so it diagonalisable in \mathbb{R}^{2}." Does it just mean that M_{1}=M_{1}^{T} or something else...
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