Questions about symmetry, in group theory, geometry or elsewhere in mathematics. See https://en.wikipedia.org/wiki/Symmetry

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Symmetric Linear Transformation Matrix Multiplication

Assume that S is a matrix in $ \mathbb R^{nxn} $. Prove that $(S \vec x)^T S \vec y = \vec x \cdot \vec y $ I understand that if S is a symmetric matrix, then the transpose of S equals S. I don't ...
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3answers
62 views

If $A$ and $B$ are arbitrary $n \times n$ matrices, prove that $(A^TB^TBA)$ is symmetric

My attempt: $(A^TB^TBA)^T$=$(A^T)^T(B^T)^TB^TA^T$=$(AB)B^TA^T$ $\ne$ $(A^TB^TBA)$ therefore $(A^TB^TBA)$ is not symmetric.
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1answer
17 views

Find a basis of the space V of all symmetric 3x3 matricies, and thus determine the dimension of V

I need help finding the general element or matrix of $V$. Do I need to find the basis of the nullspace and basis of the image to solve this problem?
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0answers
17 views

Identifying Symmetric Matricies

If $A$ and $B$ are arbitrary $nxn$ matrices, which of the matrices must be symmetric? $something^T$ means the transpose of "something". $A^T$$A$ $A-A^T$ $A^TB^TBA$
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2answers
44 views

What is the dimension of a certain vector subspace of invariant matrices?

$\newcommand{\Sig}{\Sigma}$ Let $\Sig$ be a diagonal matrix with strictly positive entries on the diagonal. Define $V=\{B \in M_n\mid B\Sig +\Sig B^T=\Sig B +B^T \Sig \}$ (where $M_n$ is the vector ...
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1answer
19 views

Does Linear Transformation transforms orthonormal bases of symmetric matrix into orthogonal vectors?

How would you show it? I went all over the book: looked at Linear Transformation definition again, looked at orthogonality of bases for symmetric matrices. I got to the point: Since L is linear ...
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2answers
231 views

Reflection with respect to a parabola

I know how to find a reflection with respect to one of the axis or with respect to the origin, but let's say I want to find the reflection with respect to a parabola, how do I do it? Let's say we have ...
2
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36 views
+50

Matrix representations of the generators of the full octahedral group

I want to find matrix representations of the generators of the full octahedral group which has the presentation $\{a,b,c|a^2=1,b^3=1,(ab)^4=1,ac=ca,bc=cb\} $ where a,b and c are the generators of the ...
3
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1answer
19 views

Combination of even and odd functions

Can someone help me how to show that any function $f(x)$ defined on a symmetrically placed interval can be written as a sum of an even and a odd function? What is the special role played by ...
0
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1answer
38 views

Showing a function is odd

I have this equation: $$ f(x) = \frac{2x^2+3}{x-2} $$ and I have to prove it has half-turn symmetry around the point (2,8). I know that for a function to have half-turn symmetry, it needs to have ...
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13 views

Symmetric and reflexive closure on positive integers

Finding the symmetric and reflexive closure of the relation $R = \{(a, b) | a > b\}$ on the set of positive integers. For symmetric closure I have: $R = \{(a, b) | a > b\} U \{(b, a) | a > ...
3
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1answer
42 views

What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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0answers
11 views

Determining if Poset based on Domain and Comparison Operator?

Can someone help me with how to think about the below problem? I know that a poset is a relation which is reflexive, antisymmetric, and transitive, but unless I'm dealing with finite sets I have a lot ...
0
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1answer
32 views

Proof concerning preservation of negative eigenvalues under matrix addition of symmetric matrices

I've been working on this proof from a test review for about an hour and a half trying to figure out what to do. I have also scoured the internet in an attempt to find a similar problem to hint at ...
3
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0answers
24 views

Fixed-point subspace of $O(2)^-$, a subgroup of $O(3)$

$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane. The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition ...
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0answers
11 views

Translational Invariance

Consider a system of objects labeled by the index $I$, each object located at position $x_{I}$. (For simplicity, we can consider one spatial dimension, or just ignore an index labeling the different ...
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16 views

Symmetry of Fourier Transform Spectrum

Given the continuous Fourier Transform $X(f)=\int_{-\infty}^{\infty}x(t)e^{-j2\pi ft}dt$ A sufficient condition for it to exist is $E_{x}=\int_{-\infty}^{\infty}|x(t)|^{2}dt < +\infty$. We can ...
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1answer
24 views

Symmetric Matrices and Diagonalization

Hi, I am trying to figure this problem out, but I am having difficulty. What I do know is that since A is symmetric, then S must be orthogonal. Also that S^(-1) must equal S^t (Transpose). ...
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27 views

Metrizability of space of unitary operators on Hilbert space

Let $\mathbb{H}$ be a (complex) Hilbert space. Define $\mathbb{P}$ as a projective space over $\mathbb{H}$, i.e. $\mathbb{P}=\left(\mathbb{H}\setminus\{0\}\right)/\sim$, where $f\sim g$ iff there ...
3
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1answer
51 views

Characterizing orthogonally-invariant norms on the space of matrices

Denote by $M_n$ the space of $n \times n$ real matrices. We say a norm on $M_n$ is orthogonal invariant if: $$\|OX \|=\| XO\|=\|X \| \, \, \forall O \in O_n,X \in M_n$$ I am trying to characterize ...
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2answers
64 views

All-Russian Olympiad question (sum of symmetrical functions)

(All-Russian Olympiad, $1995$, $11^{th}$ Graders, Final Round) Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions each of which has a vertical ...
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1answer
43 views

Commutation relation between su(N) and clifford algebra generators.

Why does the $ \gamma_5 $ matrix commute with the generators of the $su(N)$ algebra? In the case of the chiral symmetry from physics, [$Q_a$, $Q_b^5$] = $i \epsilon_{abc}Q^5_c$ where the $Q_a, Q^5_a$ ...
3
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0answers
22 views

Isometries of the canonical left invariant metric on $GL_n$

Consider $GL_n(\mathbb{R})$ with the left-invariant Riemannian metric obtained by left translating the standard metric on $T_IGL_n \cong M_n \cong \mathbb{R}^{n^2}$. (i.e for $X,Y \in T_IGL_n \,: ...
2
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1answer
20 views

Can the isometry group of a metric space determine the metric?

Let $(X,d)$ be a metric space. There are always other metrics on $X$ which generates the same topology, and have the same isometry groups, for instance $\tilde d =\sqrt d$. (The same will be true for ...
2
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0answers
48 views

A brief answer for the determinant of a matrix

I am given the following matrix $A=(a_{ij})_{6 \times 6}$, where $a_{ij}=\sum_{k=1}^{10} x_k^{i+j-2}$. Remark: If $A=(a_{ij})_{10 \times 10}$ with the same $a_{ij}$ defined above, the answer is very ...
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1answer
58 views

a*b = a/b = b/a (what's this symmetry called?)

I was playing around with numbers the other day, and I found an interesting symmetry, that I would like to know if it has any specific name assigned to it. Let's assume the notation n:a to refer to ...
3
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5answers
101 views

Fastest way to show that $D_6 \to S_5$ is an injective homomorphism

I want to show that there is an injective homomorphism from $D_6 \to S_5$ where $D_6$ denotes the dihidral group of order 12 and $S_5$ the symmetric group. But I'm not sure how I can do this ...
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3answers
67 views

Does $y=(ax^{2}+bx+c)/(dx+e)$ have any lines of symmetry? [closed]

Does $y = \dfrac{ax^2 + bx + c}{dx + e}$ have any lines of symmetry. If it does, what are they, and if not, how would one prove that it doesn't have any lines of symmetry? (Please consider the ...
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1answer
30 views

Does $y=2/x$ have any lines of symmetry?

Lines of symmetry for $y=1/x$ are $y=x$ and $y=-x$. Does $y=2/x$ likewise have lines of symmetry?
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1answer
16 views

Surjective mapping of matrices under rotational and reflection symmetries

Let me preface this by saying that I'm not a mathematician and that I'm having a hard time stating my problem in the proper terms. Nevertheless, I'm faced with a problem for which I think an elegant ...
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1answer
28 views

Show if lines $L_1$ and $L_2$ are parallel then reflection $r_{L_1L_2}$ is a translation

I need to show that if lines $L_1$ and $L_2$ are parallel then the reflection $r_{L_1L_2}$ is a translation. I can draw two lines that are parallel and show how a reflection in both results in a ...
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2answers
21 views

necessary and sufficient conditions under which a symmetric matrix X

How to answer this question ? Provide necessary and sufficient conditions under which a symmetric matrix $X$ can be written as $X =A^T A$ for some matrix $A$.
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1answer
36 views

“Minimising” a linear combination of orthonormal vectors

I am working with two $24\times1$ orthonormal vectors, and I wish to find the linear combination of these for which the maximum possible number of entries in the resultant vector are equal to (or ...
3
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1answer
45 views

Analysis of coefficients of $x^k+\dfrac1{x^k}$ polynomials

Given $x+\dfrac1x=n$, I derived several expressions in terms of $n$ to solve for $x^k+\dfrac1{x^k}$ and put them in a chart as shown below. My questions is how are the coefficients of these ...
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1answer
52 views

Show that $(x'Ay)x=(xx'A)y$

Suppose $x,y$ are vectors and $A$ is a symmetric invertible matrix. Show that $(x'Ay)x=(xx'A)y$. How can one prove the above? I am aware that matrix multiplication is not necessarily associative. ...
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0answers
37 views

How to use symmetry in triple integrals/ in $\mathbb{R}^3$

Hi I think I still get confused on one aspect of calculating triple integrals. I am used to utilizing symmetry in $\mathbb{R}^{2}$ , but I think there is something major I am missing in ...
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1answer
34 views

Why, given an object with rotational symmetry, is the axis of symmetry a principal axis?

When consulting textbooks and notes online about principle axes of inertia, I couldn't find a source which directly addressed the reasoning/proof behind following statement: "Given an object with a ...
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1answer
61 views

When does a linear combination of trigonometric functions have an axis of symmetry?

I am trying to find out when a linear combination of $\sin(ax)$ and $\cos(bx)$ has an axis of symmetry. Clearly, $\sin(x)+\cos(x)$ has an axis of symmetry at $\pi/4$. It seems as if $\sin(3 ...
3
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0answers
73 views

Can a L-shaped figure be divided into 5 congruent shapes?

Given a square, remove one quarter of it. Can the resulting L-shaped figure be divided into 5 congruent shapes? If not, how can we prove that fact? I tried using circles, triangles and smaller ...
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1answer
50 views

Resources for solving Euclidean geometry problems using symmetries

I know a number of books that treat geometry from the viewpoint of transformations/symmetries. However, very few of them actually teach someone to solve Euclidean geometry problems using said ...
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1answer
41 views

Use only Symmetry in question related to triangle to prove $\angle KPQ = \angle PKL$

In figure $\triangle APB$ , $\triangle QBC$ & $\triangle ARC$ are isosceles. $PK$ $\perp AC$, also $Ql$ $\perp AC$ $P,B,L$ are co linear. Also $\angle APB =\angle BQC = \angle ARC = 120^o$. How ...
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2answers
95 views

Show that $H = \bigcup_{n > 0} S_n$ is not equal to Sym$(X)$ for $X = \{1,2,3,4,…\}$.

Let $X = \{1,2,3...\}$ be the set of positive natural numbers, $S_n$ the permutation group, and Sym$(X)$ the set of all bijections from X to X with operation composition. I have the following ...
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28 views

Easiest way to prove a congruency

Let's take ABCD a convex quadrilateral, such that the diagonals AC and BD are perpendicular, and the angles ABC and CDA are equal. What is the easiest way to prove that the triangles ABC and CDA are ...
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Symmetry in graphs of polynomials

I am learning graph skectching. Its well known that quadratic polynomials over reals are symmetric about their minima/maxima. But today I discovered an interesting result that Cubic polynomials are ...
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1answer
41 views

Group generated by two reflections order 2

I have two reflections $r$, $s$ with $r^{2}=s^{2}=e$. Also $rs \neq sr$ and $(rs)^4 = (sr)^4= e$. I know $r$ and $s$ generate a group of order $16$ but that was by writing each matrix multiplication ...
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1answer
21 views

Anti-symmetric ways

A car dealer lines up his best objects for sale. He has 'n' Porsche and 'n' Ferrari. How many anti-symmetric ways are there to arrange these cars? (Anti-symmetric means that if ith from left is a ...
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1answer
68 views

finding irrational positive eigenvalues of a real symmetric $3 \times 3$ matrix

I have a real symmetric $3 \times 3$ matrix and I know all its eigenvalues are positive and irrational. All I care about are the eigenvalues (don't need the eigenvectors). What is the most efficient ...
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1answer
36 views

Symmetries of octahedron with $2$-faces action

Want to do the $2$-faces action. We use the Orbit stabilizer theorem. Let $X$ be the set of faces (any face can go to any face), $X=\{1,2,3,4,5,6,7,8 \}$. Where $1,2,3,4$ are the front faces of the ...
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Why is $J_n$ not symmetric, for $n\notin\mathbb Z$, while Bessel's equation is still symmetric?

Bessel's equation, $$x^2y''+xy'+(x^2-n^2)y=0,$$ has even parity, regardless of the value of $n$. So a solution of this equation must be even or odd. However, the Bessel functions $J_n$, which are ...
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Multivariable Calculus - Symmetry in integrals, cancelling out

So, I've got the integral $\iiint_R$$(xy+z^2)dV$ over the set $0\le z \le 1-|x|-|y|$. So, I've gotten that this makes out a pyramid in $\mathbb{R}^3$ with corners in $(0,0,1)$, $(0,1,0)$ and ...