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Questions tagged [projection]

This tag is for questions relating to "Projection", which is nothing but the shadow cast by an object. An everyday example of a projection is the casting of shadows onto a plane. Projection has many application in various areas of Mathematics (such as Euclidean geometry, linear algebra, topology, category theory, set theory etc.) as well as Physics.

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If $A\subseteq B$, then $B^{\perp} \subseteq \Pi(B\mid A^{\perp})$

Suppose A, B are closed linear subspace of Hilbert space, I want to show if $A\subseteq B$, then $B^{\perp} \subseteq \Pi(B\mid A^{\perp})$, where $\Pi$ is the projection operator. I'm not sure how to ...
maskeran's user avatar
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3 votes
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$L^2$ vs $L^\infty$ projection

Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is $$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$ where $d_\...
Davide Maran's user avatar
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2 answers
71 views

How to project one circle onto another for the purposes of angles?

Two circles are concentric with the same radius. Circle 2 (blue) is tilted relative to the xy plane. They coincide at some nodal axis (dotted line) which is the x axis. Now we advance to some point A ...
DrZ214's user avatar
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-1 votes
0 answers
20 views

Finding minimal injective projection [closed]

Suppose there is a set of axes $S = \{X_0, \ldots , X_n\}$ and a set of points $P = \{p_0, \ldots , p_m \}$; A projection $f: S \stackrel{f}{\longmapsto} s$ is called minimal if $s$ has the fewest ...
multus's user avatar
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5 votes
1 answer
60 views

$(Px,x)=\|Px\|^2$ implies that $P$ is a projection.

Let $H$ be a Hilbert space over $\mathbb{R}$ and $P$ is a bounded linear operator over it. If $$ (Px,x)=\|Px\|^2,\quad \forall x\in H, $$ I want to show $P$ is a projection. If $H$ is a Hilbert space ...
MakaBaka's user avatar
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0 answers
12 views

Bergman projection maps $L^q \left (\mathbb D^2 \right )$ boundedly onto $\mathbb A^q \left (\mathbb D^2 \right )$ for any $q \geq 2.$

Let $\mathbb A^2 \left (\mathbb D^2 \right )$ be the Bergman space consisting of square integrable holomorphic functions on $\mathbb D^2$ and $\mathbb P : L^2 \left (\mathbb D^2 \right ) \...
Anacardium's user avatar
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1 answer
23 views

Orthogonal Projection onto a Polyhedron (Matrix Inequality)

How to efficiently solve: $$\begin{align*} \arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} \\ \text{subject to} \quad & \begin{aligned}...
Royi's user avatar
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1 answer
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Characteristic polynomial of an orthogonal projection

Q: What is the characteristic polynomial of an orthogonal projection onto a (two-dimensional) plane through the origin in $\mathbb R^4$? Ans: $x^2(x-1)^2$ Can someone please explain how to do this ...
Jason Xu's user avatar
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1 vote
2 answers
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Orthogonal Projection onto the Convex Hull of Permutation Matrix

Given a matrix $\boldsymbol{Y} \in \mathbb{R}^{m \times n}$ where $n \leq m$. I want to find its projection onto the Convex Hull of Permutation Matrices: $$ \mathcal{P} = \left\{ \boldsymbol{P} \mid \...
Royi's user avatar
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0 answers
32 views

Proving that cylinder-to-sphere projection is area preserving.

I was going through the textbook Computer Graphics: Principles and Practice, and in the chapter for light, it talks about how the projection from a cylinder to sphere is area-preserving. I was trying ...
kre 1's user avatar
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0 answers
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Shadow a Cube with body diagonal along y axis

We have a cube with its body diagonal (greatest diagonal joining two opposite vertices) aligned along the y axis. Thus, the cube stands on one vertex with the opposite vertex directly above it. The ...
BlueInfinite1729's user avatar
3 votes
0 answers
48 views

Further decomposition of isotypic components in a representation

Let $(V,\rho)$ be an orthogonal (resp. unitary) representation of finite group $G$ whose irreducible representations over the same field as $V$ are $W_i$ with character $\chi_i$. We have $V \cong \...
khashayar's user avatar
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2 votes
1 answer
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Name and properties of $\sup_{a \in A}\| P_K(a)-a\|_H$ where $P_K$ is the projection onto a closed convex set $K$

Let $K$ be a closed convex subset of a Hilbert space $H$ and $A \subset H$ be some other set. Does the following quantity $$\sup_{a \in A}\| P_K(a)-a\|_H$$ where $P_K\colon H \to K$ is the orthogonal ...
math_guy's user avatar
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Canonical epimorphisms vs Projections vs Idempotents

In the context of linear algebra, a projection $P$ from $V\cong \mathbb{R}^n$ onto $S \cong \mathbb{R}^m$ is an $n \times n$ idempotent matrix (i.e. $P^2=P$) such that image of $P$ is $S$. In a more ...
khashayar's user avatar
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1 vote
1 answer
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Operator less than a projection with equal trace?

If $A$ is an operator on a Hilbert space $H$ and $P$ is a projection on $H$ with $0 \leq A \leq P$ and $\operatorname{Tr}(A) = \operatorname{Tr}(P)$, then $A = P$. My proof: Since $0 \leq A \leq P$, ...
Mara Jade's user avatar
1 vote
1 answer
45 views

Complemented subspace of set o functions whose integral is zero

If we consider the space $L^1([0,1])$, and its subspace: $$ Y = \left \{ f \in L^1([0,1]) : \int_{[0,1]} f d \mu = 0 \right \}$$ What would be the complemented subspace of Y, i.e. set Z such that: $$ ...
AlaskaYoung's user avatar
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55 views

Criterion for products of projections.

Given an arbitrary vector space $V$ and two (linear) projections $P$ and $Q$. In the case of orthogonal projections, it is relatively easy to show that the product $PQ$ is a projection if and only if $...
Konstruktor's user avatar
1 vote
0 answers
39 views

Projection of real representations onto the isotypic components

Let $(V, \rho)$ be a representation of a finite group $G$ over field $\mathbb{F}$ and $W_i$ be irreducible representations (irreps) of $G$ over $\mathbb{F}$ with dimension $d_i$ and character $\chi_i$....
khashayar's user avatar
  • 2,285
1 vote
0 answers
71 views

Riez sequences and frames of subspaces, provided an orthonormal basis

Given a Hilbert space $\mathcal H$ and a sequence $\lbrace f_n\rbrace\subset\mathcal H$, we say that it is a frame for $\mathcal H$ if there are constants $A,B>0$ such that $A\|f\|^2\leq \sum |\...
confusedTurtle's user avatar
1 vote
1 answer
28 views

Complete quadrangle $ABCD$; line $r$

Consider the complete quadrangle $ABCD$ and a line $r$ that doesn't pass through any of these points. $X= r \cap AB; Y= r \cap AC; Z=r \cap AD; T=r \cap BC; U=r \cap BD, V= R \cap CD$ and $X',Y',Z',T'...
J P's user avatar
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0 answers
23 views

What is the intuition to create an orthogonal design matrix in an iterative way (without Gram-Schmidt)?

Let $X_1, X_2, \dots, X_n$ be $n$ observations between $[-1,1]$ and $X_i \ne X_j$ if $i\ne j$. Let $\phi_0(x)=1$, $\phi_1(x) = > 2(x-a_1)\phi_0(x)$. When $r\ge 1$, $\phi_{r+1}(x)=2(x-a_{r+1})\...
Kaven Lin's user avatar
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20 views

Projection of vectors from starting basis onto orthogonal complement

Gram-Schmidt process allows us to produce a basis $\{w_1,...,w_n\}$ starting from a basis $\{v_1,...,v_n\}$. If I define $W_j$ to be the subspace generated by $\{w_1,...,w_j\}$ for $j=1,..,n.$ Can I ...
user926356's user avatar
  • 1,482
1 vote
1 answer
18 views

On point-polygon distance

Consider a simple (i.e. non self-intersecting) polygon $P$, not necessarily convex, identified by the vertices $V_1,\dots,V_n \in \mathbb{R}^2$. Let $y\in\mathbb{R}^2$ be a given point. Consider the ...
matteogost's user avatar
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Linear Algebra: Orthogonal basis to find proj of $\mathbf{w}$ onto $\mathbf{y}$?

I'm currently studying for my final exam for linear algebra, and I'm a bit confused about how to find the projection of $\mathbf{w}$ onto $\mathbf{y}$. I already found the orthogonal basis for $W$, ...
ejry's user avatar
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3 votes
0 answers
33 views

Find the $L_2$ projection on the set $S$

Let $f\in L_2[0,1]$ and $S = \{h\in L_2[0,1]:\; \int h = 1,h\geq 0\}$. What is the $L_2$ projection of $f$ on $S$? I guess that it is not available in closed-form? I would guess that the projection is ...
John Smith's user avatar
0 votes
1 answer
59 views

Orthogonality of two projections in $\mathcal B(H)$

Let $H$ be a Hilbert space and $\mathcal B(H)$ denotes the space of all bounded operators on $H$. For any projections $p,q \in \mathcal B(H)$ we define a projection from $H$ onto the closure of linear ...
DenOfZero's user avatar
  • 127
0 votes
0 answers
30 views

Find the matrix with respect to the canonical basis of the orthogonal projection of $R^3$ on subspace $S$

Find the matrix with respect to the canonical basis of the orthogonal projection of $R^3$ on subspace $S=\text{span}{(1,0, -1),(0,1 ,-1)}$. I have to get the orthogonal subspace ($S^⊥$), and that is ...
ssj's user avatar
  • 21
0 votes
0 answers
42 views

Non-Euclidean projections

Background Let $y\in\mathbb{R}^2$ be a given point and let $V_1,V_2\in\mathbb{R^2}$ be the vertices of a given segment. Define the projection of $y$ over the segment $s=(V_1,V_2)$ as \begin{equation*} ...
matteogost's user avatar
0 votes
1 answer
19 views

Find position vector in spherical coordinates given Fx, Fy, Fz, Mx, My, Mz, and r [closed]

I've run into a bit of a roadblock figuring out this problem. I need to find the position vector given all three dimensions of force and moment as well as the radius of the sphere. I've gotten the ...
Jake1234's user avatar
0 votes
1 answer
24 views

Calculating/enumerating the euclidean projection between two spheres

Let $P \subset \mathbb{R}^n$ be a finite set of $n$-dimensional euclidean points; $r_p \in \mathbb{R}^{+}$ be the radius of the sphere centered at $p \in P$; And $c_p = \{ x \in \mathbb{R}^n : \|x - ...
Matheus Diógenes Andrade's user avatar
3 votes
0 answers
59 views

proof: projecting a projective variety onto the first component remains a projective variety

I have recently begun studying algebraic geometry and have decided to start with Harris' classic, supplementing it with these online notes from a lecture, based on his book, given by Harris himself at ...
mathrandom's user avatar
0 votes
0 answers
13 views

Contractive Projections: Formation of Proper Subspaces and Implications in Complemented Subspace Theory

I'm a student delving into functional analysis, and I came across Lemma 2.1.1 "Topics in Banach Spaces" in my studies. It talks about normalized block basic sequences in $l^{p}$, stating ...
Cauchy's user avatar
  • 23
0 votes
1 answer
47 views

Does this projection make sense?

Given $x\in\mathbb{R}^d$, we let $\pi_i(x)$ be the $i$-th coordinate of $x$ in the basis of eigenvectors $(v_1,\dots,v_d)$; in other words, $x=\sum^d_{i=1}\pi_i(x)v_i$. Source: PÒLYA URNS AND OTHER ...
Dada's user avatar
  • 701
2 votes
1 answer
40 views

$s(e+f) = e\lor f$ for projections $p,q\in B(H)$

Let $H$ be a Hilbert space and $p,q\in B(H)$ orthogonal projections, i.e. $$p=p^*= p^2, \quad q=q^* = q^2.$$ I want to show that $s(p+q) = p\lor q$ where $s(p+q)$ is the support projection of $p+q$ ...
Andromeda's user avatar
  • 840
2 votes
1 answer
93 views

Coordinate-wise projections separate points in the closed span of a sequence

Let $X$ be a Banach space over $\mathbb K$ where $\mathbb K=\mathbb R$ or $\mathbb K=\mathbb C$, $(x_n)$ a linearly independent sequence (can be normalized with necessary or convenient) in $X$, $E$ ...
Emerick's user avatar
  • 189
1 vote
0 answers
33 views

Rank of sum of two projection operators if their difference is also a projection

Suppose $E_1, E_2$ are two projection operators on a finite-dimensional vector space $V$ over a field $F$ of characteristic $\neq 2$ such that $E_1-E_2$ is also a projection. I could deduce that $E_1 ...
Naba Kumar Bhattacharya's user avatar
1 vote
0 answers
22 views

Time series projection problem, the stationarity is unnecessary here?

Problem (Time-Methods-Peter-J-Brockwell problem 2.8): Suppose $\{X_t,t=1,2,\cdots\}$ is a stationary process with mean zero. Show that $$ P_{\bar{sp}\{1,X_1,X_2,\cdots,X_n\}}X_{n+1}=P_{\bar{sp}\{X_1,...
onsdriver's user avatar
3 votes
0 answers
241 views

Degenerate spectrum for the sum of projections onto unit vectors that sum to zero implies a symmetry of the vectors?

I am interested in a sum of projections $A = \sum_{i=1}^N \mathbf{a}_i\,\mathbf{a}_i^T$, where $\mathbf{a}_i$ are real column vectors with unit norm and $\sum_i \mathbf{a}_i = \mathbf{0}$. So $A$ is a ...
Luzveraz's user avatar
0 votes
1 answer
47 views

Prove P²=P for orthogonal projection formula

Given the following orthogonal projection formula $\hat{x}_w$ of $x \in \mathbb{R}^n$ on the vector $\frac{w}{\|w\|} \in \mathbb{R}^n$ defined by $\hat{x}_w := \frac{c \cdot w}{\|w\|}$ where $c = \...
fearloathing121's user avatar
0 votes
1 answer
27 views

Orthogonal Projection in an Enlarged Hilbert Space

Let $(H, \langle \cdot, \cdot \rangle_H)$ and $(U,\langle \cdot, \cdot \rangle_U )$ be Hilbert Spaces such that $H$ embeds into $U$. Let $M$ be a closed subspace of $U$, and define $\mathcal{P}$ to be ...
RiaDoog's user avatar
1 vote
1 answer
38 views

Orthogonal projection of a double-sided cone onto a plane

I would like to solve the following problem: a double cone with vertex at the origin of coordinates (0, 0, 0) is given (See fig. 1). The blue cone is symmetrical to the yellow one w.r.t. the origin; ...
Gino's user avatar
  • 372
0 votes
2 answers
42 views

Orthographic Projection and Concentric Circles [closed]

Let $C$ be the bounded set of concentric circles centered at the origin. Let $r_n = \sqrt{\frac{n}{\pi}}$ for any integer $n \in [1, k]$ be the radius of the circle $C_n \in C$. Assume $C$ is a ...
Aphrontos's user avatar
  • 115
0 votes
0 answers
30 views

Determine if point can be projected on any segment in polyline

I am trying to implement a script that determines if a series of points can be projected on any segment of a polyline. The projection must be exactly on the segment, not in the rest of the line ...
NeuroTheGreat's user avatar
0 votes
0 answers
45 views

sum of projected area and a generalization of Oppenheim's inequality

From this post: In $\mathbb R^4$, let $U$ be a 2D plane, let $\pi_1$ be the projection from $U$ onto $xy$-plane and $\pi_2$ be the projection from $U$ onto $zw$-plane, then $\det\pi_1+\det\pi_2\le1$, ...
hbghlyj's user avatar
  • 3,045
0 votes
0 answers
39 views

Projections onto orthocomplement 2D planes in 4D

From this post, It is possible to go another route and generalize. The orthogonal projection $\pi_2:U\to\Pi_2$ may be restricted to $\Pi_1$, and it has a "hypervolume distortion factor" $\...
hbghlyj's user avatar
  • 3,045
0 votes
0 answers
19 views

I'm not able to create propper essential matrices and I don't know why

I'm currently trying to generate an essential matrix from two cameras with known parameters. According to Wikipedia this can easily be done by calculating the transformation ($R$, $\vec{t}$) from one ...
RobinW's user avatar
  • 77
0 votes
1 answer
40 views

The projection of the polynomial onto U parallel to V

Prove that the space $P_3$ of polynomials of degree no higher than $3$ is the direct sum of subspaces $U$ and $V$ and find the projection of the polynomial $t^3$ onto $U$ parallel to $V$, where \begin{...
Alice P.'s user avatar
1 vote
0 answers
31 views

lengths of orthogonal projections of the standard basis on a subspace

Let $e_1,\ldots,e_n$ be the standard basis in $\mathbb{R}^n$. Suppose the scalars $\lambda_1,\ldots,\lambda_n$ satisfy $0< \lambda_1,\ldots,\lambda_n\leq1$ and $\lambda_1^2+\ldots+\lambda_n^2 = m$, ...
Ayden Chang's user avatar
0 votes
1 answer
29 views

projection of vector not equal length of the projection

I have 2 vectors, a and b, from the origin to a plane. The normal vector to the plane, is n. (a dot n) and (b dot n) is the same, and I learned it is equal to the projection of a onto n (and b onto n)....
trogne's user avatar
  • 129
1 vote
1 answer
37 views

About the boundary of the $x$-section of a bounded domain

Decompose $\mathbb{R}^N = \mathbb{R}^{N_1} \times \mathbb{R}^{N_2}$, let $\Omega \subset \mathbb{R}^N$ be a bounded domain and take $x$ in the set $$\Pi_1(\Omega) = \{ x \in \mathbb{R}^{N_1} : (x,y) \...
Lucas Linhares's user avatar

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