Reflection is a transformation that fixes a line or plane or a more general subset. Reflections appear in geometry, linear algebra, complex analysis, differential equations, etc -- therefore, this tag must be used with a tag describing the area of mathematics.

learn more… | top users | synonyms

2
votes
1answer
19 views

Is $\operatorname{Stab}(\lambda)$ generated by the simple reflections it contains, for $\lambda\in A_0$?

For a finite Weyl group, the stabilizer of an element in the fundamental domain is generated by the simple reflections of the Weyl group that is contains. Does the same still hold for the closure of ...
2
votes
1answer
45 views

Proof of Isometry and Reflection

$ V = \mathbb R^n$ is provided by the standard scalar product and by the standard basis $S$. $ W \subseteq V $ is a vector subspace and $ W^\bot$ is its orthogonal complement. a) Prove that there ...
0
votes
0answers
14 views

Positive roots expressible as a combination of simple roots whose reflections pairwise commute?

Suppose you have a subset $\alpha_1,\dots,\alpha_r$ of simple roots in a reflection group whose corresponding reflections $\{s_1,\dots,s_r\}$ pairwise commute. Why are the only positive roots ...
2
votes
2answers
94 views

Functional equation $P(X)=P(1-X)$ for polynomials

I have encountered the following problem : Find all polynomials $P$ such as $P(X)=P(1-X)$ on $\mathbb{C}$ and then $\mathbb{R}$. I have found that on $\mathbb{C}$ such polynomials have an even ...
1
vote
1answer
31 views

Proof of a theorem on Reflection Groups

I am reading the book Finite Reflection Groups by Grove and Benson. I didn't understand the following proof. See $(a_1,t)$. What is $t$ here? Then Why the inequality $(r,t)-2(r,r_{i_1})(r_{i_1},t)&...
2
votes
2answers
41 views

What is the difference between and projection and a reflection, in vector transformation?

In my text book I have the problems of finding the standard matrix of the given linear transformation from $\mathbb{R}^2$ to $\mathbb{R}^2$; Projection onto the line $y = -x$. Reflection in the line ...
0
votes
0answers
20 views

Construction of a triangle using symmetry

I need to construct a triangle $\Delta \textrm{ABC}$ knowing that $t_a = AS$, $|AS| = 6\, cm$, $|\measuredangle \textrm{BCA}| = 30°$ and $|AB| = 5.5 \,cm$. I've been told that it's possible to do it ...
1
vote
1answer
20 views

Why is reflection length equal to codimension of fixed subspace in a real reflection group?

If you have a finite, real reflection group, why can the length function $\ell(-)$ be interpreted as the codimension of the fixed subspace, or alternatively, as the number of eigenvalues different ...
1
vote
0answers
29 views

Why are the conjugated generating reflections the only reflections of a finite reflection group?

Why are the conjugated generating reflections the only reflections of a finite reflection group? Suppose $W$ is a finite reflection group. (i.e $W$ is finite and is generated by a set of orthogonal ...
0
votes
0answers
12 views

Determining the roots corresponding to the Cartan matrix of a graph

Let $G$ be a graph with no loops or multiple edges, with nodes labelled $1, \dots, n$. Define the Cartan matrix $C$ of $G$ to be the $n \times n$ matrix with $$ C_{ij} = \begin{cases} 2 & \mbox{...
0
votes
0answers
13 views

Compute Green's function for Helmholtz equation in a square

Consider the following Dirichlet problem for the Helmholtz equation: $$ (\Delta + k^2)u = \delta(x, y), \qquad (x,y) \in [-a/2, a/2]\times[-a/2, a/2]\\ u|_{x=\pm a/2} = u|_{y=\pm a/2} = 0. $$ If I ...
0
votes
1answer
36 views

What is the red angle if the green and blue angles are known and the yellow angles are equal in the following image?

What is the red angle if the green and blue angles are known and the yellow angles are equal in the following image: Edit: What's the formula for figuring out the red angle?
1
vote
0answers
14 views

Reflection principle for the modulus of the Brownian Motion

I have the following question. Suppose we define $M(t)=\sup_{0\le s\le t}|B(s)|$, where $B$ is an ordinary Brownian motion in $\mathbb{R}$. How can we compute $P(M(t)\ge a)$? Is it $2P(|B(t)|\ge a)$? ...
0
votes
0answers
11 views

Clarification on a reflexive function

$R_1 = \{(a, b) | a ≤ b\}$ is a reflexive function, but I'm confused on why it is. $a≤b$ but doesn't that not necessarily mean that there is an $a$ that will equal $b$? Couldn't all of the $a$ very ...
2
votes
2answers
24 views

The expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$.

Using vectors I tried obtain the expression for reflection of a ray line $ax+by+c=0$ reflected by a mirror whose normal is given by $a'x+b'y+c'=0$. The point of intersection is $$X=\frac{bc'-cb'}{ab'-...
0
votes
0answers
24 views

Reflection Principle in multi-variable complex functions

I was wondering if the reflection principle: $\bar{f(x)}=f(\bar{x})$, covers also the multivariable case, e.g: $\bar{F(z,w)}=F(\bar{z},\bar{w})$ and if yes, to which extent. More presicely, in the ...
1
vote
0answers
42 views

Find a vector V that is parallel to the reflection of a matrix about a line.

Specific Question: Let $M=(1/53)⋅[[45, 28], [28,-45]]$. M is the matrix of the reflection about a line. Find a vector v of magnitude (53)^0.5 parallel to that line. Currently in class we are ...
0
votes
1answer
53 views

Reflect point across line with matrix

What is the transformation matrix that I multiply a point by if I want to reflect that point across a line that goes through the origin in terms of the angle between the line and the x-axis? In other ...
1
vote
2answers
36 views

Inverse of a reflection matrix?

What does it mean when I say that the inverse of a reflection matrix is the reflection matrix itself? What does it mean, intuitively or geometrically, to invert a reflection matrix? Explanation with ...
5
votes
1answer
58 views

Relation between reflection group and coxeter group

Reflection group is defined see https://en.wikipedia.org/wiki/Reflection_group. An abstract Coxter group is defined to have generators $s_1$, $s_2$, ..., $s_n$ and relations $s^2_i=e$, $(s_is_j)^{m_{...
1
vote
1answer
43 views

Composition of two reflections (non-parallel lines) is a rotation

I am trying to prove that the composition of two reflections in non-parallel lines (i.e. lines that intersect) is a rotation. From observation I can see that using $L_1$ as the $x$-axis and $L_2$ as ...
2
votes
3answers
96 views

Determining whether an orthogonal matrix represents a rotation or reflection

The exercise asks us to determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of ...
0
votes
3answers
64 views

Give the line of reflection or angle of rotation of an orthogonal 2x2 matrix

Determine whether the given orthogonal matrix represents a rotation or a reflection. If it is a rotation, give the angle of rotation; if it is a reflection, give the line of reflection. \begin{...
1
vote
1answer
43 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 ...
1
vote
1answer
22 views

Isometric transformation

If $S$ belongs to the line $p$, prove that each of the isometrics $\rho_{S,\omega}$ ◦ $ \sigma_p$ and $ \sigma_p$ ◦ $\rho_{S,\omega}$ should be axial reflection. Can anyone please help me with this ...
0
votes
0answers
18 views

Realizing a reflection group as a Weyl group

Question 1. Suppose given a compact, connected Lie group $G$ and a subtorus $S$ (not maximal) such that the effective image $N$ of the $\mathrm{Ad}$-action of the normalizer $N_G(S)$ on the Lie ...
4
votes
1answer
23 views

Method of determining symmetries in an irregular polygon (2D or 3D)?

Thank you in advance for helping. Given a polygon with $n$ vertices, $$P = \begin{bmatrix} x_{1} & x_{2} & ... & x_{n} \\ y_{1} & y_{2} & ... & y_{n} \end{bmatrix}$$ how does ...
0
votes
0answers
24 views

Characteristics of $\vec u$ in equation of the reflection of $\vec x$ about the line $N$

The linear transformation of the reflection of $\vec x$ about line $N$ is $$\vec x = 2 \text{proj}_N(\vec x)\vec x - \vec x= 2(\vec x \cdot \vec u) \vec x- \vec x.$$ Is the unit vector, $\vec u$, ...
1
vote
0answers
49 views

Range of a standard brownian motion, using reflection principle

With a standard brownian motion $B_t$, I'm trying to find the distribution of the "range": $$R_{t} = \sup_{0 \leq s \leq t} B_s - \inf_{0 \leq s \leq t} B_s = \overline{M_t}-\underline{M_t}$$ The ...
3
votes
2answers
114 views

How to determine the reflection point on an ellipse

Here is my problem. There are two points P and Q outside an ellipse, where the coordinates of the P and Q are known. The shape of the ellipse is also known. A ray comming from point A is reflected by ...
1
vote
3answers
43 views

Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: $(x,...
2
votes
1answer
41 views

Finding the glide reflection using a compass or straightedge

Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge. ...
0
votes
1answer
26 views

Possible eigenvalues of projection and reflection operators

What are the eigenvalues of an (orthogonal) projection operator that projects vectors onto some hyperplane passing through the origin? Well, for vectors $v$ orthogonal to this hyperplane, the ...
2
votes
2answers
66 views

Given $a,b \in S^n$, then there exists an isometry $f: S^n \rightarrow S^n$ such that $f(a) = b$

$S^n = \{x\in \mathbb{R}^{n+1} : \|x\| = 1\}.$ I am using this definition: isometry is a surjective function $f:M \rightarrow N$ between two metric spaces $(M,d)$ and $(N,\rho)$ such that $$\rho (f(x)...
0
votes
0answers
49 views

Isometries of reflections

Suppose $R_\rho \in M_2$ denotes a reflection across a line which is through the origin and at a anticlockwise angle of $\rho$ with the x-axis. The question asks that for any $n \in \mathbb{N}$, does ...
0
votes
2answers
26 views

Find Normal from Reflection

I have a vector A with direction $\vec D$. I also have another vector B with direction $\vec R$. (think of $\vec R$ to be the reflection of $\vec D$ at point B.) From the reflection formula:$$ \vec ...
0
votes
0answers
37 views

Linear algebra reflection or rotation

I have a matrix $$\begin{bmatrix}2&-\frac12\\1&1\end{bmatrix}.$$ How to define matrix is rotation or reflection?
2
votes
1answer
34 views

$s$ is a reflection of the plane about $x=1$. $r$ is a reflection across the x-axis.

$s$ is a reflection of the plane about the vertical line $x=1$. $r$ is a reflection across the x-axis. $g$ is an isometry and $grg^{-1}=s$. What is $g$?
2
votes
2answers
60 views

Express rotation of the plane about a point as the product of a translation and a rotation about the origin

I need help with the following problem: a) Let $s$ be a rotation of the plane with angle $\frac{\pi}{2}$ about the point $(1,1)^t$. Write the formula for $s$ as a product $t_a \rho _\theta $ Edit: $...
1
vote
1answer
29 views

Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ \...
1
vote
1answer
61 views

Vector Reflection Matrix over line

I need to find a matrix that reflects vectors over any line y= mx + b. Furthermore, I need to be able to find the components of both of these vectors. Thanks!
0
votes
0answers
21 views

Regarding the Schwarz Reflection Principle, is getting the analyticity of f(z) on the real axis a consequence of the theorem itself,

or a consequence of Morera's Theorem? Basically, I want to be able to cite it correctly, e.g., can I say we have not only continuity of f(z) along $R$ (by assumption) but also it turns out that $f(z)$...
-2
votes
1answer
31 views

let X has a finite dimensional show that X is reflexive.

We know If a normed space X is reflexive, then $X'$ is reflexive.and also Reflexive normed spaces are Banach. but can you proof if X has a finite dimensional Then X is reflexive.
1
vote
1answer
24 views

Householder QR problem

Can somebody give me a hint or help me to solve this problem. Let V be a p×q matrix with orthonormal columns (p > q), and $M = I−2VV^T$ , with I being the p × p identity matrix. The matrix M can ...
0
votes
0answers
101 views

Find the Inverse Matrix of a Transformation

Let $ f: \mathbb R^3 \rightarrow \mathbb R^3$ be a linear mapping which reflects $\bar{x}$ over the plane $x_1+x_2+x_3 = 0$ . You are given the standard matrix for $f$ is: $$ \frac{1}{3}\begin{...
0
votes
1answer
32 views

Roots of height 1 are necessarily simple.

Suppose $\Phi$ is a root system, $\Pi \subset \Phi$ is a fundamental system (let $\Pi = \{r_1,...,r_l\}$). Now any root $r \in \Phi$ is a linear combination of the elements of $\Pi$ with all the ...
3
votes
0answers
30 views

Reflection through a hyperplane, orthogonality propoerty

While I was studying for my exam, I found Stanford's EE263 course's old homework questions and this particular one attracted my attention, however couldn't solve it. If you can help me, I will really ...
0
votes
1answer
183 views

Find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c

I want to find a 3x3 Matrix that reflects in an Arbitrary Line y=mx+c I have a matrix with me but it doesnt take into account the '$c$' This is the matrix \begin{pmatrix} \frac{1-m^2}{m^2+1}& ...
1
vote
2answers
152 views

Finding the reflection that reflects in an arbitrary line y=mx+b

How can I find the reflection that reflects in an arbitrary line, $y=mx+b$ I've examples where it's $y=mx$ without taking in the factor of $b$ But I want to know how you can take in the factor of $b$...
0
votes
1answer
18 views

$PQRS$ is a convex quadrilateral.Find the area of $PQRS.$

Let $P$ be the point $(3,2).$Let $Q$ be the reflection of $P$ about the $x-axis.$Let $R$ be the reflection of $Q$ about the line $y=-x$ and let $S$ be the reflection of $R$ through the origin.$PQRS$ ...