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.

Filter by
Sorted by
Tagged with
0 votes
1 answer
27 views

Spectral theorem for diagonalizable matrices and associated spectral projectors

I was going through my lecture notes and was wondering about the connection between the diagonalized form of a matrix and it's spectral projectors: Let $ A \in M_{n\times n}(F) $ with spectrum $\sigma(...
okoci's user avatar
  • 1
1 vote
1 answer
72 views

"Agreeing" orthogonal projections

This is a projection excercise I'm stuck on. Lets say we have two planes in R3, $A$ and $B$, which both go through the origin. We also have two orthogonal projections, $ProjA$ and $ProjB$, projecting ...
hellothere's user avatar
0 votes
0 answers
24 views

Is it possible to compute the projection of a linear system solution on a constraint set instead of computing the solution of the constrained problem?

I try to solve the following problem for a personal project : $M$ is a wide matrix, $M \in \mathbb{R}^{m\times n}$, with $m << n$, such that $MM^t = D$, a diagonal matrix, and $A$ is a square, ...
Baptiste GENEST's user avatar
2 votes
0 answers
33 views

How does Jacobi form lives projectively on the torus $\mathbb{C}/(\mathbb{Z}+\tau \mathbb{Z})$?

I saw the statement in the question from the book Moonshine Beyond the Monster. We are given the definition : a group hom $\rho : G \rightarrow PGL(V)$ is called a projective representation. I can't ...
Mahammad Yusifov's user avatar
0 votes
1 answer
51 views

Technical name of a specific subset

Consider a set $A$ in $\mathbb{R}^3$. Let $(x,y,z)$ denote a representative element of $\mathbb{R}^3$. Let $$ A(x=3)\equiv\{(x,y,z)\in A: x=3\}. $$ Does this set has a specific technical name? For ...
Star's user avatar
  • 250
0 votes
0 answers
30 views

Why is $proj_{Ox} \overrightarrow{M_1M_2} = |{\overrightarrow{M_1M_2}}| \cos \varphi$ true?

The following image comes from the text and is a good visual aid for the descriptions to follow. Background The book defines the projection of a directed segment in this way: The projection $proj_{...
Paul Ash's user avatar
  • 981
3 votes
2 answers
64 views

If $P$ is a selfadjoint operator on a Hilbert space satisfying $P^4 = P$, is $P$ an orthogonal projection?

I’ve been trying to solve this problem to no avail. I have proved already that the spectrum is a subset of $\{0,1\}$ just like for orthogonal projections. I’ve also proved $\lVert P \rVert = 1$. ...
Neckverse Herdman's user avatar
0 votes
1 answer
41 views

Prove that if $p$ and $q$ are projections with the same kernel, then $p\circ q=p$ and $q\circ p=q$

Let $K$ be a field, and let $E$ be $K$-vector space. Let $p$ and $q$ be two endomorphisms of $E$. Prove the following proposition: ($p$ and $q$ are projections, and $\ker{p}$ = $\ker{q}$) $\...
virtualcode's user avatar
0 votes
0 answers
15 views

Eigendecomposition of a real symmetric matrix obtained by "projection"

I have a real square matrix $s\times s$ that writes $$ M = PU\Lambda U^\top P^\top $$ with $P = \left( \mathrm{I}_s ,0\right)\in \mathbb{R}^{s\times r}$, $U \in \mathbb{R}^{r\times r}$ orthogonal, and ...
Piou42's user avatar
  • 1
0 votes
0 answers
22 views

Projective transformation between colinear points in 2D with 2x2 homography possible?

Referring to: 2.5 The projective geometry of 1D by Richard Hartley and Andrew Zissermann. Can I compute a projective transformation matrix 2x2 from two sets of 3 1D colinear points? I want to find the ...
Pj Toopmuch's user avatar
0 votes
0 answers
22 views

From convergence of orthogonal projection to orthogonal series expansion in reproducing kernel Hilbert spaces.

Introduction: Let $\mathcal{H}$ be a Hilbert space of functions $\Omega\to\mathbb{R}$ with reproducing Kernel $K:\Omega\times\Omega\to\mathbb{R},\,\Omega\subset\mathbb{R}^d,\, d>1$, where $K$ is ...
Max Stuthmann's user avatar
0 votes
0 answers
37 views

Projection on a convex set that defined by variables not showing in the objective

The projection problem I want to solve is to find a vector $u\in R^d$ so that $\min_{(u,z)}: \|| u'-u\||^2_2$, where $u_i\leq z_i, z \in \Omega$. Here $\Omega$ is a convex set. As shown all $z_i$ are ...
E.J.'s user avatar
  • 939
1 vote
0 answers
28 views

Reducing the size of the product of a matrix and a vector

I have a matrix $P$ with size $(d,d)$ and two vectors $x$ and $y$ of size $(d,1)$. I want to reduce the size of the quantity $(Px-Py)$ by setting the size of $P$ to be $(p,d)$ where $p \leq d$ to ...
Ki Chao's user avatar
  • 43
-1 votes
1 answer
54 views

Write vector as the sum of w and w orthogonal [closed]

Let $\mathbb{R}^3$ be given with the standard inner product and let $W$ be the subspace spanned by $\left( \begin{array}{c} 4\\ -2\\ 4\\ \end{array} \right)$ and $\left( \begin{array}{c} -2\\ 6\\ 2\\ \...
zeze's user avatar
  • 9
2 votes
1 answer
177 views

Determining the relative dimensions of a right circular conical frustum from a perspective image

Suppose you have a right circular conical frustum with bottom base radius $r_1$ and top base radius $r_2$, and height $h$. Now you take a (perspective) image of this frustum. The camera used to ...
of course's user avatar
  • 21.3k
1 vote
2 answers
33 views

Determining the coordinates of the vertices of a triangle from two images taken by two cameras with known position/orientation

Suppose you have a triangle with unknown side lengths hanging in space with unknown position and orientation (i.e. nothing about the dimensions or position or orientation of the triangle is known). ...
of course's user avatar
  • 21.3k
1 vote
1 answer
73 views

Determining the coordinates of the vertices of a triangle from its camera image

Suppose you have a pinhole camera (no lens) with known position and orientation with respect to the world. Now with this camera, you take an image of a triangle that has known side lengths of ...
of course's user avatar
  • 21.3k
0 votes
2 answers
133 views

Projection on the orthocomplement of a finite-dim Hilbert space

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space. Moreover, let $\mathcal{P}_2 \subset \mathcal{H}$ and $\mathcal{P}_3 \subset \mathcal{H}$ be a finite-dimensional and an infinite-...
Vuk's user avatar
  • 319
2 votes
1 answer
56 views

Are the angles from the diagonal of a square to each side always equal? Even after rotation? And perspective?

If you draw a square on a sheet of paper, and draw the diagonal, are the apparent angles from the diagonal to each side equal, even if you rotate the sheet in 3D space? Visual Example Transcript of ...
Ali Stokes's user avatar
1 vote
1 answer
88 views

Sort rectangles by size, but with a tunable advantage favoring squareness

I want to sort some rectangles from biggest to smallest, but with a tunable advantage favoring squarer rectangles. I'm sure this is super easy but I'm not getting it. I think it's related to topics ...
Jason Kleban's user avatar
1 vote
0 answers
14 views

Smooth parameterization of a 3d unit vector using 2 variables [duplicate]

Let $\vec{x}(a,b)$ be a parameterization of an arbitrary unit vector in 3 dimensions: $a,b\in\mathbb{R}$ $||\vec{x}(a,b)||=1$ for all $a,b$. For a given unit vector, there exists some finite $a,b$ ...
devtk's user avatar
  • 111
3 votes
2 answers
68 views

sum of projections equivalence over vector space

Let $V$ be a vector space, let $p, q$ projections over $V$. I am trying to prove the statement which says that if $p+q$ is a projection, then $p\circ q=q\circ p=0$. I could only show that $p\circ q + ...
User666x's user avatar
  • 844
1 vote
2 answers
58 views

Is $\langle f - m, m \rangle = 0$ sufficient for $m$ to be an orthogonal projection?

I was looking at https://en.wikipedia.org/wiki/Hilbert_projection_theorem and I wondered about the following statement, but I wasn't able to prove or disprove it. Suppose that $M$ is a subspace of a ...
Harry Partridge's user avatar
4 votes
2 answers
139 views

Projection with same rank

Let $H$ be a Hilbert space (over $\mathbb{R}$ or $\mathbb{C}$), and $P$, $Q$ are two projections over $H$ such that $\|P-Q\|<1$. I am searching for a simple proof for $$ \dim\operatorname{Ran}P=\...
MakaBaka's user avatar
  • 175
1 vote
1 answer
53 views

Finding an explicit formula for the projection on a subspace of $\ell^2$

Consider $\ell^2(\Bbb C)=\{x=(x_n)_{n\in \Bbb N}|\, \sum_{n\ge 1}|x_n|^2<+\infty\}$ endowed with the usual inner product and the subspace $V:=\{x=(x_n)_{n\in \Bbb N}\in \ell^2 |\, x_1=2x_2-x_3\}$. ...
Sine of the Time's user avatar
1 vote
1 answer
42 views

What do the parts of the Gram-Schmidt process mean and represent in space?

I am struggling to understand what the different parts of the Gram-Schmidt process represent. Suppose we have a basis $\{x_1, x_2\}$ We would then find a orthogonal basis by doing the following : $$...
Yassine's user avatar
  • 11
1 vote
1 answer
51 views

How do projective transformation matrices work?

I've been working on a tool to turn photos of graphs into digital data (I know there are already tools out there, I wanted to understand the mechanism and I find learning by doing the easiest), but I ...
Laurengineer's user avatar
1 vote
1 answer
34 views

Are range and kernel of a partial isometry disjoint if the isometry generates a central projection?

Given a von Neumann algebra $M$ on an Hilbert space $H$, my naive understanding is that if I have a partial isometry $v\in M$ and projection $p\in P(M)$ such that $v^* v = p$, where $v^*$ is the ...
DerHutmacher's user avatar
5 votes
1 answer
104 views

Finding an explicit formula for the projection on a subspace of an Hilbert space

Given the Hilbert space $L^2([-1,1])$ endowed with the usual inner product, consider the following operator: $$Tf:=\int_{-1}^1f(x)e^x \mathrm dx $$ Let $N:=\ker T$, find an explicit formula to find ...
Sine of the Time's user avatar
0 votes
0 answers
31 views

Commuting Bounded linear projections

Let $X$ be a Banach (or a Hilbert) space, $T,P\in L(X)$ are bounded linear operators such that $P^{2}=P$ and the operator $TP-PT$ has a finite rank. I want to construct another projection $Q\in L(X)$ (...
Luffy's user avatar
  • 45
1 vote
3 answers
82 views

How do I project a point $P$ onto a sphere from a projective point $C$ that is NOT located at the sphere's center or surface?

I have a sphere of radius $1$ centered on the point $(0, 0, 1)$. I also have a point $C = (0, 0, H)$ where $0 < H < 1$ from which to project objects onto the sphere. It is located inside the ...
Lawton's user avatar
  • 1,673
1 vote
0 answers
20 views

How to show that a projecting an n-dimensional simplex decrease its volume?

Let $[v_{0},v_{1},\cdots,v_{n}]$ be an $n-$simplex in $n$ dimensional space. Let $H$ be the projection of $v_{0}$ onto the hyperplane spanned by $\{v_{1},\cdots ,v_{n}\}$. Prove that $|[v_{0},v_{1},\...
OneLamp's user avatar
  • 339
0 votes
0 answers
39 views

Is there really no simple form for the projection onto the positive part of the hyperboloid model?

Consider the "positive part" of the hyperboloid model $$ \mathbb H_+^d := \{ (x_1, \ldots, x_d, x_{d + 1}) \in \mathbb R^{d} \times \mathbb R_{> 0}: \sum_{k = 1}^{d} x_k^2 = x_{d + 1}^2 - ...
ViktorStein's user avatar
  • 4,788
2 votes
2 answers
59 views

Projections from endomorphism algebra of $G$-equivariant maps

Let $(V,\rho)$ be a representation of finite group $G$. $V$ can be decomposed to the direct sum of isotypic components $$V \cong V_1 \oplus V_2 \oplus \cdots \oplus V_m,$$ where, for each $i$, $V_i$ ...
khashayar's user avatar
  • 2,174
0 votes
0 answers
40 views

What does it mean if the change of volume of a linear transformation is -1?

Let $P:\Bbb{R}^2\to\Bbb{R}^2$ be projection onto the line with equation $x + 2y = 0$. Find $\det(P)$. When computing the previous problem, I find that the change of volume is $-1$, however, the ...
lubindarryl's user avatar
0 votes
0 answers
11 views

Is the greedy projection the best (entropy-minimizing) projection?

Let $Y,X_1,X_2$ be binary random variables. I'm training a naive binary tree classifier and I need to find the best binary projection $f: \{0,1\}^2 \to \{0,1\}$ such that $H(Y|f(X_1,X_2))$ is minimal. ...
user35443's user avatar
  • 373
1 vote
1 answer
30 views

Fibre of linear projection from an irreducible hypersurface

Let $H \simeq \mathbb{P}^n$ be a hyperplane in $\mathbb{P}^{n+1}$, and $p$ a point not lying on $H$. For every point $x \in \mathbb{P}^{n + 1} \setminus \{p\}$, the line $\overline{px}$ in $\mathbb{P}^...
Lasting Howling's user avatar
0 votes
2 answers
46 views

In $\Bbb R^2,$ let $L$ be the line $y = mx.$ Find an expression for $T(x, y),$ where $T$ is the projection on $L$ along the line perpendicular to $L.$

In $\Bbb R^2,$ let $L$ be the line $y = mx$, where $m\neq 0$. Find an expression for $T(x, y),$ where $T$ is the projection on $L$ along the line perpendicular to $L.$ (See the definition of ...
Thomas Finley's user avatar
0 votes
0 answers
27 views

Do the projection errors resulting from linear projections onto each other satisfy any equality restriction?

Suppose I have two random variables $X_1$ and $X_2$ that are not independent of each other. Consider the linear projections of $X_1X_2$ and $X_1$ onto each other: $L(X_1X_2|X_1)$ and $L(X_1|X_1X_2)$, ...
ExcitedSnail's user avatar
0 votes
0 answers
30 views

Projections in semi-normed spaces

I have a question regarding the projection onto a finite-dimensional subspace of a semi-normed vector space: Let $V$ be a real vector space (either finite or infinite-dimensional) and let $\langle\...
huhu27's user avatar
  • 3
0 votes
0 answers
9 views

Existence of projection $P$ equivalent to $P' \circ T \circ P''$ for projections $P', P''$ and smooth translation $T$?

A projection in the linear algebraic sense is a linear map $P$ such that $P^2 = P$. I'm interested in knowing when there is guaranteed to exist a projection $P$ such that $P = P' \circ T \circ P''$, ...
Tanishq Kumar's user avatar
2 votes
0 answers
89 views

How can I get the appropriate coordinates to display 3D-hyperbolic space in the Beltrami Klein model?

I am trying to create a "game" with a hyperbolic world. The goal is to have some objects (for example trees, cars, buildings, cubes, spheres, ...) which are displayed using the Beltrami ...
juwa's user avatar
  • 29
4 votes
1 answer
266 views

Finding the missing centre or vertex of a projected rectangle.

In the attached image BCDE is a rectangular object that is projected through focal point J to produce a perspective image with vertices B'C'D'E' on the vertical blue plane. The plane of the rectangle ...
KDP's user avatar
  • 1,079
-1 votes
1 answer
82 views

If $P$ is an idempotent and $\langle Px,x\rangle \geq 0$, show that $P$ is a projection

If $P$ is an idempotent on a Hilbert space $H$ and $\langle Px,x\rangle\geq 0$ for all $x\in H$, I’m trying to prove that P is a projection, i.e., $\text{null}P=(\text{ran}P)^{\bot}$. I have proved ...
OSCAR's user avatar
  • 571
0 votes
1 answer
16 views

Orthogonal Projection onto the Set of Square Matrices with a Unit Trace

The problem is given by: $$ \begin{align*} \arg \min_{\boldsymbol{x}} \quad & \frac{1}{2} {\left\| \boldsymbol{X} - \boldsymbol{B} \right\|}_{F}^{2}, \; \boldsymbol{B} \in \mathbb{R}^{n \times n} \...
Royi's user avatar
  • 8,671
1 vote
2 answers
144 views

Why can the area projected onto a plane be calculated with the dot product of the vector area and the unit vector of the plane?

I was trying to find the projection of a flat surface onto an arbitrary plane and I came across this Wikipedia article, The projected area onto a plane is given by the dot product of the vector area $...
Invenietis's user avatar
0 votes
0 answers
224 views

An example of an non-invertible operator F such that a truncated operator F will be invertible

We have: $X$ - any Banach space $F : X \to X$ (linear bounded and non-invertible) $P_n : X \to X$ Where $P_n$ is projector that strongly converges to the identity operator $I$ as $n \to\infty$ Can ...
TorteDeline's user avatar
0 votes
0 answers
33 views

Projecting a $v$ onto $v+\epsilon$.

Suppose $v \in \mathbb{R}^n$, $\|v\|=1$, and $v^*=v+\epsilon$ be a noise corrupted version, with $\epsilon \in \mathbb{R}^n$ a random vector with entries Gaussian(0,1). Can I derive an expression for ...
user310374's user avatar
0 votes
2 answers
47 views

A doubt in solving a projection mapping and orthogonality question

Let $l^2= \{ (x_1,x_2,x_3,\dots):x_n \in \mathbb{R} \text{ for all } n \in \mathbb{N} \text{ and } \sum_{n=1}^{\infty} x_n^2 < \infty \}.$ For a sequence $(x_1,x_2,x_3,\dots) \in l^2,$ define $$\...
MathRookie2204's user avatar
5 votes
1 answer
563 views

Can one compute the location of the unseen point?

My question is quite simple. I have two images, on the first one I know the location of points $P1, P2, P3$, and $P4$. In the second image, I know the location of $P2'$, $P3'$, $P4'$, and point $Q'$. ...
apraglez's user avatar

1
2 3 4 5
28