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.

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What algebra is generated by $\mathrm{O}(2)$?

The unit complex numbers can be identified with the $2 \times 2$ special orthogonal matrices $\mathrm{SO}(2)$. The problem with $\mathrm{SO}(2)$, however, is that its not closed under $\mathbb{R}$-...
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Reflection principle for walk possible steps right, left and stay

I need to use reflection principle for one dimensional walk with equally possible steps right, left and stay. I would like to know if hold a similar identity to that of question Is there an intuitive ...
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Minimal polynomial of projection on a plane?

If $g: \mathbb{R}^3 \to \mathbb{R}^3$ is the projection on a plane, what is the minimum polynomial of g? Related to this what is the minimum polynomial of a reflection with respect to a plane?
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Affine and linear reflections

Let $\gamma$ - affine reflection in complex space, which is transformation with properties: (1) $\gamma$ is a motion (thus linear part of $\gamma$ : $\mathbf{Lin} \gamma \in U(V)$), (2) $\gamma$ ...
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Holomorphic extension using reflection principle

I cannot really understand how the reflection principle works. I have found various forms of the theorem, but the notes i use are not so clear on how we actually extend functions and what is the ...
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1answer
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irreducible unitary reflection group

Let $G$ be a finite irreducible unitary reflection group (i.e. without G-invariant subspaces). Given orthonomal basis, we have that $g_1 \in GL(V)$ commutes with every element of $G$. It is said that ...
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Uniqueness of reflection stabilizing a finite generating set [duplicate]

Let $V$ be a finite dimensional real vector space, $v\in V$ . Define a reflection relative to $v$ to be a linear transformation which sends $v$ to $-v$ and fixes pointwise a subspace of ...
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How to turn the reflection about $y=x$ into a rotation.

If we reflect $(x,y)$ about $y=x$ then we get $(y,x)$. And because $x^2+y^2=y^2+x^2$ this can also be represented by a rotation. Using this we get: $$(x,y)•(y,x)=2xy=(x^2+y^2)\cos (\theta)$$ Hence ...
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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 ...
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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 ...
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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 ...
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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 ...
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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)&...
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1answer
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Why is this linear transformation uniquely determined?

Let $V$ be an $n$ dimensional real vector space. Let $S = \{ v_1, ... , v_m \}$ be a spanning set for $V$ not containing the zero vector. Suppose that $\phi_i: V \rightarrow V, i = 1, 2$ are linear ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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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{...
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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 ...
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1answer
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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?
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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 $2P(|B(t)|\ge a)$? ...
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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 ...
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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'-...
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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 ...
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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 ...
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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 ...
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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 ...
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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_{...
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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 ...
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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 ...
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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{...
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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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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 ...
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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 ...
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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 ...
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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, $\vec u$, ...
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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 ...
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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 ...
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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,...
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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. ...
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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 ...
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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)...
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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 ...
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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 ...
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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?
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1answer
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$s$ 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 $g$?
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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: $...
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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 \ \...
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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!