# 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. ### Condition such that the symmetric matrix has only positive eigenvalues

My attempt: $$\begin{vmatrix} 1-\lambda & b\\ b & a-\lambda \end{vmatrix} =0$$ $$(1-\lambda)(a-\lambda)-b^2=0$$ $$a-\lambda-a\lambda+\lambda^2-b^2=0$$ $$\lambda^2+(-1-a)\lambda +a-b^2=0$$ The value of ##\lambda## will be positive if D < 0, so $$(-1-a)^2-4(a-b^2)<0$$ $$1+2a+a^2-4a+4b^2<0$$...
2. ### A Varying an action wrt a symmetric and traceless tensor

Consider a Lagrangian, #L#, which is a function of, as well as other fields #\psi_i#, a traceless and symmetric tensor denoted by #f^{uv}#, so that #L=L(f^{uv})#, the associated action is #\int L(f^{uv}, \psi_i)d^4x #. To vary w.r.t #f^{uv}# , I write...
3. ### A Regular vs stable orbits in spherically symmetric potentials

I am struggling with Hamiltonian formulation of classical mechanics. I think I have grasped the idea of canonical transformations, including the idea of angle-action variables and invariant tori in phase space. Still, few points seem to elude my understanding... Let's talk about a particle...
4. ### I Solving Spherically Symmetric Static Star Equations of Motion

Hi guys, I can't seem to be able to get to $$(\rho + p) \frac {d\Phi} {dr} = - \frac {dp} {dr}$$ from $$T^{\alpha\beta}_{\,\,\,\,;\beta} = 0$$ the only one of these 4 equations (in the case of a spherically symmetric static star) that does not identically vanish is that for ##\alpha=r##...
5. ### I Congruence for Symmetric and non-Symmetric Matrices for Quadratic Form

I learned that for a bilinear form/square form the following theorem holds: matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form. Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...
6. ### I Error propagation and symmetric errors

Hello! I am a bit confused about how to interpret symmetric error when doing error propagation. For example, if I have ##E = \frac{mv^2}{2}##, and I do error propagation I get ##\frac{dE}{E} = 2\frac{dv}{v}##. Which implies that if I have v being normally distributed, and hence having a...
7. ### Can every symmetric matrix be a matrix of inertia?

Hello, I am often designing math exams for students of engineering. What I ask is the following: Can I choose any real 3x3 symmetric matrix with positive eigenvalues as a realistic matrix of inertia? Possibly, there are secret connections between the off-diagonal elements (if not zero)...
8. ### I Determinant of a specific, symmetric Toeplitz matrix

Let us define matrix ##\mathbf{B}_n=[b_{ij}]_{n\times n}## as follows $$[b_{ij}]_{n\times n}:=\begin{cases} b_{ij} = \alpha\,,\quad j=i\\ b_{ij}=\beta\,,\quad j=i\pm1\\ b_{ij}=1\,,\quad \text{else}\end{cases}\,,$$ where ##\alpha\,,\beta\in\mathbb{R}## and ##n\geq2##. ##\mathbf{B}_4##, for...
9. ### A Reason out the cross product (for the moment): a skew symmetric form

I am sure you are all familiar with the cross product in 3D space. i cross into j gives k. Cyclic Negative, if reversed, etc. I am sure you are all familiar with the definition as: norm of the first vector, norm of the second, sine of the angle, perpendicular (but direction using right hand...
10. ### A Why is this matrix symmetric here?

Goldstein 3rd Ed, pg 339 "In large classes of problems, it happens that ##L_{2}## is a quadratic function of the generalized velocities and ##L_{1}## is a linear function of the same variables with the following specific functional dependencies: ##L\left(q_{i}, \dot{q}_{i}, t\right)=L_{0}(q...
11. ### B Spatial curvature around a spherically symmetric mass

Suppose I measure the circumference of a circular orbit round a massive object and find it to be c. Suppose I then move to a slightly higher orbit an extra radial distance δr as measured locally. If space was flat I would expect the new circumference to be c + 2πδr. Will the actual measurement...
12. N

### Find a and b to make this equation symmetric about the y-axis: y = ax^2 + bx^3

My friend asked for help with this precalculus question. I could not help him. So, I decided to post here. Find a and b when the graph of y = ax^2 + bx^3 is symmetric with respect to (a) the y-axis and (b) the origin. (There are many correct answers.) I don't even know where to begin.
13. ### I Un-skewing a skew symmetric matrix (for want of a better phrase)

Hello Say I have a column of components v = (x, y, z). I can create a skew symmetric matrix: M = [0, -z, y; z, 0; -x; -y, x, 0] I can also go the other way and convert the skew symmetric matrix into a column of components. Silly question now... I have, in the past, referred to this as...
14. ### Finding the directions of eigenvectors symmetric eigenvalue problem

In the symmetric eigenvalue problem, Kv=w^2*v where K~=M−1/2KM−1/2, where K and M are the stiffness and mass matrices respectively. The vectors v are the eigenvectors of the matrix K~ which are calculated as in the example below. How do you find the directions of the eigenvectors? The negatives...
15. ### Spherically symmetric states in the hydrogen atom

The equation $$\frac{\hbar^2}{2m}\frac{d^2u}{dr^2}-\frac{Ze^2}{r}u=Eu$$ gives the schrodinger equation for the spherically symmetric functions ##u=r\psi## for a hydrogen-like atom. In this equation, substitute an assumed solution of the form ##u(r)=(Ar+Br^2)e^{-br}## and hence find the values...
16. ### A Radial acceleration for spherically symmetric systems in GR

Hello, I was reading few papers discussing modified gravity theories and their use in understanding galaxies with no dark matter by checking for anomalous velocity dispersion. Now, the author was using 4 gravity theories MOND, Weyl, MOG and Emergent gravity. The thing is he had provided the...
17. ### Showing that real symmetric matrices are diagonalizable

Summary:: Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix and let ##\lambda \in \Bbb R## be an eigenvalue of ##A##. Prove that the geometric multiplicity ##g(\lambda)## of ##A## equals its algebraic multiplicity ##a(\lambda)##. Let ##A \in \Bbb R^{n \times n}## be a symmetric matrix...
18. ### I Are all processes CPT symmetric like measurement, stimulated emission?

https://en.wikipedia.org/wiki/CPT_symmetry says "CPT theorem says that CPT symmetry holds for all physical phenomena" - e.g. we could imagine decomposition of given phenomena into Feynman diagrams and apply CPT symmetry to all of them. However, for some o processes such reversibility seems...
19. ### B How to prove the associative law of symmetric difference?

I'm trying to prove the associative law of symmetric difference (AΔ(BΔc) = (AΔB)ΔC ) with other relations of sets. A naive way is to compare the truth table of two sides. However, I think the symmetric difference is not a basic one, it is constructed form other relations, that is AΔB =...
20. ### I Solving the Spherically Symmetric Einstein Equation

Can be Einstein equation rewrited into some simpler form, when suppose only spherically symmetric (but not necessarily stationary) distribution of mass-energy ? If yes, is there some source to learn more about it ? Thank you. edit: by simpler form I mean something with rather expressed...
21. ### I Const Curvature Scalar & 3-Torus: Is It Maximally Symmetric?

Spatial slices of the Robertson-Walker metrics are maximally symmetric so they must have a constant curvature. Is it true that in three Riemannian dimensions that a constant curvature scalar determines whether the volume is finite or infinite? Carroll seems to have given a counter-example for...
22. ### I What is the definition of trace for n-indexed tensor in group theory?

I was reading zee's group theory in a nutshell. I understand that we can decompose a 2 index tensor for rotation group into an antisymmetric vector(3), symmetric traceless tensor(5) and a scalar(trace of the tensor). Because "trace is invariant" it put a condition on the transformation of...
23. ### I Subgroup axioms for a symmetric group

Hi, The textbook asks me to use subgroup axioms to prove why a set of permutations that interchange two specific symbols in S4 is or isn't a subgroup of the symmetric group, and the same for a set of permutations that fix two elements. My guess is that the set of permutations that interchange...
24. ### Why does a symmetric wavefunction imply the angular momentum is even?

I looked in the instructor solutions, which are given by: But I don't quite understand the solution, so I hope you can help me understand it. First. Why do we even know we are working with wavefunctions with the quantum numbers n,l,m? Don't we only get these quantum numbers if the particles...
25. ### I Spherically Symmetric Metric: Is Singularity Free?

Is there a spherically symmetric metric that doesn't have a singularity in the middle of it(like the schwartzchild metric). Something like our planet.
26. ### Properties of symmetric magnetic field around ##Z## axis (cylinder)

I am trying to understand but without a succes why symmetric magnetic field around ##Z## axis make that ##\hat \phi## magnetic field is zero I can't understand why it physically happens and also how can I derive it mathematically? What does the word symmetric means when talking about magnetic...
27. ### I Understanding the Definition of Isotropic Spaces in Riemannian Manifolds

Why does the constraint: $$R_{ijkl}=K(g_{ik} g_{jl} - g_{il}g_{jk})$$ Imply that the resulting space is maximally symmetric? The GR book I'm using takes this relation more or less as a definition, what is the idea behind here?

29. ### A Symmetric limit in Peskin's and Schroeder's (page 655)

What is exactly the definition of symmetric limit? It's the first place in the book that I see this notation, and they don't even define what it means. How does it a differ from a simple limit or asymptotic limit? I found a few hits in google, but it doesn't seem to help...
30. ### Checking relation for reflexive, symmetric and transitive

Now, with the given set of natural numbers, we can deduce the relation ##R## to be as following $$R = \big \{ (1,6), (2,7), (3,8) \big \}$$ Now, obviously this is not a reflexive and symmetric. And I can also see that this is transitive relation. We never have ##(a,b) \in R## and ##(b,c) \in...
31. ### B Skew symmetric 1 dimension

Hi, if I have a equation like (just a random eq.) p_dot = S(omega)*p. where p = [x, y, z] is the original states, omega = [p, q, r] and S - skew symmetric. How does the equation appear if i only want a system to have the state z? do I get z_dot = -q*x + p*y. Or is the symmetric not valid so I...
32. ### MHB Solving a Quartic Polynomial with Symmetric Graph & Intercept -2

Find the equation of a quartic polynomial whose graph is symmetric about the y -axis and has local maxima at (−2,0) and (2,0) and a y -intercept of -2
33. ### I Proof of ##F## is an orthogonal projection if and only if symmetric

The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
34. ### Symmetric top with constant charge to mass ratio in a magnetic field

Setup: Let ##\hat{\mathbf{e}}_1,\hat{\mathbf{e}}_2,\hat{\mathbf{e}}_3## be the basis of the fixed frame and ##\hat{\mathbf{e}}'_1,\hat{\mathbf{e}}'_2,\hat{\mathbf{e}}'_3## be the basis of the body frame. Furthermore, let ##\phi## be the angle of rotation about the ##\hat{\mathbf{e}}_3## axis...
35. ### I Are electron bands symmetric in the reciprocal space?

Hi, in the lecture notes my professor gave us, it is stated that, due to Kramers theorem, the energy in a band must satisfy this condition: $$E(-k) = E(k)$$ But, judging from actual pictures of band structures I don't find this condition to be true. Here's a (random) picture I guess it looks...

39. ### I What is a symmetric ODE / what does it mean when an ODE is symmetric?

How can an ODE be symmetric? How would you plot an ODE to show off this property? (i.e. what would be the axes?)
40. ### Integration of traceless symmetric matrices

Hi, I stumbled upon an identity when studying tensor perturbations in cosmology. The formula states that $$\int d^2\hat{p} f(\hat{p}.\hat{q})\hat{p}_i \hat{p}_k e_{jk}(\hat{q}) = e_{ij}(\hat{q})/2 \int d^2\hat{p} f(\hat{p}.\hat{q})(1-(\hat{p}.\hat{q})^2),$$ where ##e_{ij}## is a symmetric...
41. ### I Maximally symmetric sub-manifold (2-sphere)

Again in pg. 166 eq 7.2 https://www.google.com/url?sa=t&source=web&rct=j&url=http://www.astro.caltech.edu/~george/ay21/readings/carroll-gr-textbook.pdf&ved=2ahUKEwi1gdbj3ODgAhXRWisKHXW_D-sQFjACegQIBhAB&usg=AOvVaw1YY2mM7uccdbX4nTxFgQO5 Here ##u^1=\theta,u^2=\phi## and v=r. The tangent vector on...
42. ### I Symmetric Connection: Does Torsion Vanish?

Does a symmetric connection implies that torsion vanishes?
43. ### MHB The decomposition for a symmetric positiv definite matrix is unique

Hey! :o We have the matrix \begin{equation*}A=\begin{pmatrix}1/2 & 1/5 & 1/10 & 1/17 \\ 1/5 & 1/2 & 1/5 & 1/10 \\ 1/10 & 1/5 & 1/2 & 1/5 \\ 1/17 & 1/10 & 1/5 & 1/10\end{pmatrix}\end{equation*} I have applied the Cholesky decomposition and found that $A=\tilde{L}\cdot \tilde{L}^T$ where...
44. ### I How to derive a symmetric tensor?

Let ##Q_ik## be a symetric tensor, so that ##Q_ik= \frac{m}{2} \dot x_i \dot x_j + \frac{k}{2} x_i x_j## (here k is also a sub, couldn't do it better with LaTeX). How do we derive such a tensor, with respect to time? And what could such a tensor mean in a physical sense? It really looks like the...
45. ### Electric field from spherically symmetric charge distributio

Homework Statement A spherically symmetric charge distribution produces the electric field E=(200/r)r(hat)N/C, where r is in meters. a) what is the electric field strength at 10cm? b)what is the electric flux through a 20cm diameter spherical surface that is concentric with the charge...
46. ### Are Similar Matrices' Eigenvalues the Same? Solving for Symmetric Matrices

Homework Statement Consider matrices A = [1 2;2 4] and P = [1 3;3 6]. Using B = P^-1*A*P, verify that similar matrices have the same eigenvalues. Find the eigenvectors y for B and show that x = P*y are eigenvectors of A. Homework Equations B = P^-1*A*P, x = P*y The Attempt at a Solution I...
47. ### Finding the order of element of symmetric group

Homework Statement Let ##n## be a natural number and let ##\sigma## be an element of the symmetric group ##S_n##. Show that if ##\sigma## is a product of disjoint cycles of orders ##m_1 , \dots , m_k##, then ##|\sigma|## is the least common multiple of ##m_1 , \dots , m_k##. Homework...
48. ### MHB Show that G is a subset of the symmetric group

Hey! :o Let $n\in \mathbb{N}$ and $M=\{1, 2, \ldots , n\}\subset \mathbb{N}$. Let $d:M\times M\rightarrow \mathbb{R}$ a map with the property $$\forall x, y\in M : d(x,y)=0\iff x=y$$ Let \begin{equation*}G=\{f: M\rightarrow M \mid \forall x,y\in M : d(x,y)=d\left (f(x), f(y)\right...
49. ### Symmetric bowl associated with a line element

Hi! I have the following problem I don't really know where to start from: A bowl with axial symmetry is built in flat Euclidean space ##R^3##, and has a radial profile giveb by ##z(r)##, where ##z## is the axis of symmetry and ##r## is the radial distance from the axis. What radial profile...
50. ### I Given order for every element in a symmetric group

Compute the order of each of the elements in the symmetric group ##S_4##. Is the best way to do this just to write out each element's cycle decomposition, or is there a more efficient way?