# Tagged Questions

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

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### 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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### “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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
### 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 ...