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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3
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1answer
32 views

Can it be said that $E(A_1)E(A_2) = E(A_2) E(A_1) = 0$ for $A_1 \cap A_2 = \varnothing\ $?

Let $(X,\mathcal A)$ be a measurable space. Let $\mathcal H$ be a Hilbert space and $E : \mathcal A \longrightarrow \mathscr P (\mathcal H)$ be a projection valued map. For all $x \in \mathcal H$ with ...
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2answers
18 views

Project a vector onto subspace spanned by columns of a matrix

From this question we know that if $x\in\mathbb{R}^{n\times1}$ is a vector, then the (normalized) outer product matrix $$ \frac{x x^\top}{||x||^2}\, \in \mathbb{R}^{n\times n} $$ can operate on ...
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0answers
32 views

Regarding an isometric drawing of a circle, how does this align perfectly?

In the above image, the circle is drawn as an isometric view. Why does it align well when we draw a six-pointed star? The radius of the larger arc is thrice the radius of the smaller arc. My question ...
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0answers
23 views

Conditions for conditional expectation E(u|y) to be linear function of y

I am considering a system of $y= {\Gamma} f+ u $, where both $f$ and $u$ are $r \times 1$ and $n \times 1$ random vectors with finite expectation and are independent of each other. $\Gamma$ is a $n\...
5
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1answer
28 views

Distance of a function from a subset in Hilbert spaces

Let $L^2[-1, 1]$ be the Hilbert space of real valued square integrable functions on $[-1, 1]$ equipped with the norm $$\|f\|_2=\big(\int_{-1}^1|f(x)|^2\, dx\big)^{\frac{1}{2}}.$$ Consider the subspace ...
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0answers
14 views

How is the length of the projection onto a subspace distributed?

Let $B$ be the unit sphere of an M-dimensional linear space and let $x$ be uniformly distributed on $B$. Let $A$ be an N-dimensional subspace of the said linear space (N<M). How is the projection ...
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0answers
19 views

Projection onto a set vs. KL minimizer to a set

I'm interested in the problem "find the closest probability measure in $KL$-divergence of a set of measures $Q$ to a given measure $p$." Specifically take $Q$ to be the set of product ...
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0answers
42 views

Projectivities of line $\mathbb{P_1}$ over $\mathbb{Z/2Z}$ field

Let the projective line $\mathbb{P_1}$ over the field of $\mathbb{Z/2Z}$. I'm asked to prove there are 6 projectivities between $\mathbb{P_1} \longrightarrow \mathbb{P_1}$ and giving the equations. ...
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1answer
29 views

Projection onto the sum of two subspaces.

Let $V_1,V_2\subset\mathbb{R}^n$ be two subspaces of dimension $d$. And $P_{V_1}, P_{V_2}$ are the orthogonal projections onto $V_1,V_2$ respectively. Let $V_1+V_2$ be the sum of these two subspaces ...
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0answers
55 views

perpendicular projections in negatively curved manifolds

Consider the hyperbolic plane $\mathbb{H}$ of dimension $2$ (think of it as the half-plane model). Let $c$ be a geodesic on $\mathbb{H}$ and let $z_1, z_2$ points such that $r=d(z_1,c)=d(z_2,c)$ and ...
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0answers
20 views

Location for maximum groupings of 3D points in two directions. (Optimisation)

I want to comeup with a point in the image its represented by (a) and (b) such that if I take a picture standing at that point the it presents better grouping of the items (balls)(better viewing ...
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1answer
22 views

If the weak limit of sequence of projections is again a projection then strong limit exists.

Let $\{P_n\}_{n \geq 1}$ be a sequence of orthogonal projections on a Hilbert space $\mathcal H$ such that weak limit of the sequence is again a projection $P$ i.e. $\left \langle x, P_n x \right \...
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1answer
33 views

Norms of orthogonal subspaces

Let V be a finite dimensional real inner product space and U, W subspaces of V such that U is orthogonal to W. Show that for any $v ∈ V$ $$||v||^2 ≥ ||proj U (v)||^2 + ||projW (v)||^2$$ Hello guys. I ...
2
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1answer
37 views

Prove that $A: (v_1)^\perp \to (v_1)^\perp ,\ e_j\mapsto Pv_j$'s determinant stays the same for all ONBs $e_j$, where $P$ is an orthogonal projection.

I am sorry for the scuffed title, but there are not enough characters. Here is the problem in full: Take a basis $(v_1,\dots,v_n)$ of $\mathbb{R}^n$ and $W=(v_1)^\perp$. Let $P$ be the orthogonal ...
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0answers
23 views

Projection onto a Line: why must the projection be a multiple of the vector $a$?

In the book: Linear Algebra and its application -Gilbert Strang-, when the projection $p$ of vector $b$ onto vector $a$ is explained, this is said: We want to find the projection point $p$. This point ...
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0answers
40 views

How to extract a component of integer vector / music interval? [duplicate]

(I will ask this question in musical terms, but this seems to be related to projecting integer vectors onto each other, which I'm unfamiliar with. Perhaps I'm just looking for some existing notation ...
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1answer
20 views

How to find the projection map onto a subspace along another subspace.

Consider the vector space $\mathbb{R}^3$ and the subspace $$V=\{ (x,y,z) \in \mathbb{R}^3 | x=y=z \} $$ Find the projection $T$ onto $V$ along the subspace $U$ generated by the vectors $(1,2,1),(0,1,1)...
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1answer
33 views

Projection $(I-p)=(I+p)$

Let $p$ be a projection such that the projection $(I-p)$ is invertible. Does $(I-p)=(I+p)$ hold? I proved it this way: $$(I-p)^2=(I-p)=(I^2-p^2)=(I-p)(I+p)$$ Hence, $$(I-p)^{-1}(I-p)^2= (I+p).$$ This ...
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4answers
49 views

Arc length between spaced points on circle when viewed under an inclination

Short version: Say you have a circle of radius $r$ with a set of $n$ equally spaced points on it, each with an arc length of $\Delta\theta=\frac{2\pi}{n}$ between each other. When that circle is ...
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1answer
37 views

Proving the equivalences of three statements regarding projections.

Let $\mathcal H$ be a Hilbert space and $P$ be a linear operator on $\mathcal H.$ The the following statements are equivalent $:$ $(1)$ $P^2 = P = P^*.$ $(2)$ The space $\mathcal K = \{x\ : Px = x \}$...
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0answers
23 views

Find a “common” row space and column space for a matrix, solving $\max_{P\in \mathbb O^{n\times k}} \quad \|P^TB P\|_F^2$ given $B$

I have an optimization problem as follows: $$\min_{\substack{A\in \mathbb R^{k\times k}\\P\in \mathbb O^{n\times k}}}\|B - P \cdot A\cdot P^T \|_F^2$$ where $B$ (whose rank can be thought as greater ...
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0answers
12 views

Composing projections in $\operatorname{CAT}(0)$ spaces

Let $\alpha\subset\Pi\subset M$ be geodesic $\alpha$ contained in a geodesic plane $\Pi$ in a $\operatorname{CAT}(0)$ space $M$, and for any convex geodesic subspace $X$ let $p_X:M\to X$ be the ...
2
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2answers
61 views

If $R$ is an orthogonal reflection, then $\frac{1}{2}(R + I)$ is an orthogonal projection

I consider a Hilbert space $H$ over $\mathbb{C}$, and a reflection $R$ in the set of bounded linear operators on $H$ denoted $B(H)$. I want to prove that if $R$ is an orthogonal reflection, then $\...
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1answer
35 views

Are geodesic preserved by projections in hyperbolic space?

In a the euclidean space $\mathbb{E}^n$ a line $\alpha$ is mapped to a line $\pi(\alpha)$ by an othogonal projection $\pi:\mathbb{E}^n\to P$ to some plane $P$. Projections in the hyperbolic space ...
2
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0answers
52 views

Orthogonal projection into a sparse subspace with $s$ dimension

Traditional orthogonal projection of a given point $y \in \mathbb{R}^n$ into a closed and convex set $D\in \mathbb{R}^n$ is defined as the follwing: $$ P_D(y)=\arg\min_{x \in D}||x-y||_2^2 $$ Now ...
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0answers
20 views

Equations of 3D Projections

Is anyone familiar with these equations? Please, let me know.
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1answer
42 views

Do projections of convex sets equal (up to an affine transformation) some intersection with a hyperplane?

Let $C$ be a convex subset of $\mathbb{R}^{n}$ and $C'$ its projection into a k-subspace $H\subseteq\mathbb{R}^{n}$ for $k\leq n.$ We can suppose for simplicity $p\colon\mathbb{R}^{n-k}\times\mathbb{R}...
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2answers
58 views

Why does a circle project its shadow as an upside down heart shape onto a corner instead of as an ellipse?

I was walking by a street corner and saw the attached image of a disc and its shadow. I was somewhat surprised to see an upside down heart shape instead of an ellipse projected onto the wall corner as ...
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1answer
15 views

Perpendicular Projection Linear Transformations

If a linear transformation T has matrix A. If T is a perpendicular projection onto the line y = -5x, then A has: eigenvector [1, -5] (this is meant to be 2 rows, 1 column) with eigenvalue 1 how would ...
0
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1answer
25 views

Projection Linear Transformations [closed]

Let $T: ℝ^2 \rightarrow ℝ^2$ be a projection onto the line $y = 3x.$ What would the determinant of the corresponding matrix (let's call it $A$) be?
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0answers
9 views

Calculate the field of view of a camera (draw a pyramid) get 4 edge vectors

I would like to draw a pyramid (representing the boundries of the camera). The input parameters are: Camera position Pc Camera direction Vc Camera up direction Vupc Camera horizontal viewing angle ...
1
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1answer
43 views

Infimum over $u$ of $(Mw+Pu)^TA(Mw+Pu)$ is $u=-(P^TAP)^{-1}P^TAMw$

Let $A$ be symmetric positive definite. Let $M$ and $P$ be full rank matrices such that every vector $v$ can be uniquely decomposed as $v=Mw+Pu$ and the matrix $[M,P]$ is invertible. $P(P^TAP)^{-1}P^...
0
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1answer
24 views

Hyperplane Reflection

Given the reflection $R_H(−2, 2, 2, −3) = (−4, 0, −2, 1)$, what is the reflection of $R_H(−1, −1, −1, 3)$, and how I can find it ? I tried to use $R_H = (2P_H b -I)b$, but I have no idea how to keep ...
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2answers
29 views

Is the globe distorted? [closed]

Since there is no any types of maps that do not distorted. The way we make the globe is to put the paper on a sphere. So, it should also be distorted. Am I right? and also If this is true then what ...
0
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0answers
30 views

Prove $\forall x_iR=\overline{\exists x_i\overline{R}}$

My working: Let $\forall x_iR$ and $\overline{\exists x_i\overline{R}}$ where k > 1 and 1 ≤ i ≤ k and R is relation on S. To prove $\forall x_iR=\overline{\exists x_i\overline{R}}$, we need to show ...
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0answers
10 views

Define an inner product so that the orthogonal projection is a given vector

Let $x_2\in\Bbb{R}^4$ and $x_1\in\Bbb{R}^2$ be fixed vectors. I am assuming $\Bbb{R}^4=\Bbb{R}^2\oplus\Bbb{R}^2$ and $x_1$ lies in the first factor. I need to find a necessary and sufficient condition ...
0
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1answer
36 views

Sequences of orthogonal projection in bounded Hilbert space

Let {$P_i$} $\subset$ B(H) (bounded Hilbert space) be the projections. Suppose that: If $P_1<P_2<...$ and there exists a projection P $\subset$ B(H)such that $\lim_{i \to \infty}P_i\zeta=P\zeta$...
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0answers
72 views

Projection operators onto convex subset decomposition

Suppose $H$ is a Hilbert space and there are closed convex sets $C,D \subset H$ such that every $h \in H$ can be written as $h= c + d$ where $c \in C$ and $d \in D$. Denote by $P_C$ the orthogonal ...
1
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1answer
20 views

Projecting onto the space of Upper-Triangular-ish Matrices

I want to solve the independent component analysis (ICA) problem for a single time domain channel of input. The problem is, in my formulation, the mixing matrix has a special structure. I want to know ...
2
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0answers
134 views

Calculate 3D Rectangle from 4 projected points on screen

Given 4 known projected points on the screen, I need to calculate a 3D rectangle where the 4 projected points coincide with the rectangle corners from the o ...
0
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0answers
33 views

Prove that $P = P_U$ if and only if $P$ is self-adjoint

I was reading Axler's Linear Algebra Done Right and the following question appears as exercise $11$ in chapter $7$, section A: Suppose $P \in \mathcal{L}(V)$ is such that $P^2 = P$. Prove that there ...
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1answer
12 views

Does an Orthogonal Projection map a basis of the space to a spanning set of the its subspace for a Hilbert Space? [closed]

For finite dimensional vector spaces it is quite easy to prove that a surjective linear operator $V\mapsto W$ maps a basis of $V$ to a spanning set of $W$. Is this property still true for linear ...
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0answers
21 views

Alternating Projection Algorithm for two sets with empty intersection

Given two closed convex sets $A,B\subset\mathbb{R}^n$, with $A\cap B=\varnothing$. Consider the alternating projection algorithm with arbitrary initial point $x_0$ and update rule: $y_k=P_A(x_{k-1})$ ...
2
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1answer
48 views

Sequence of Nested Projections in an arbitrary Normed Linear Space Converges to the Identity

I have seen similar questions on this site, most notably this one: Convergence of projections onto a nested sequence of subspaces of a Hilbert space, but they all include the Hilbert space assumption. ...
0
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1answer
43 views

If $Q \subseteq \mathbb{R}^{n-1}$ is the projection of the set $P \subseteq \mathbb{R}^n$, are $Q$ and $P$ the same set except for 1 dimension?

Let $P$ denote a polyhderal set of $x$ values in $\mathbb{R}^n: \{x: Ax \leq b\}$. Let $Q$ denote the projection of $P$ onto $\mathbb{R}^{n-1}$ (i.e., $Q \subseteq \mathbb{R}^{n-1}$). Do the sets $P$ ...
1
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0answers
35 views

Orthogonal decomposition of a vector

I have the following question here. Let $W$ be the subspace of $\mathbb{R}^5$ spanned by the vectors $$\{(1,2,2,4,-1),(1,2,-1,1,1),(0,0,-1,-1,1),(0,0,1,1,0) \}$$. a) Find an orthogonal basis for $W$. ...
2
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0answers
30 views

An approach to handle an orthogonal projection onto a nonconvex set

I have a composite problem in hand, which can be expressed as \begin{align} \min_{x \in \mathbb{R}^n} \ \ h(x) + I_{C}(x), \end{align} where $h(x)$ is $L$-smooth and $I_{C}(x)$ is an indicator ...
1
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1answer
28 views

Show that $\|\pi_c(y)-x\|_2^2 + \|\pi_c(y)-y\|_2^2 \leq \|x-y\|^2$, $\forall x$ in closed convex set $C$

Let $\mathcal{C}$ be the projection operator onto a closed convex set $\mathcal{C}$. Prove $\|\pi_c(y)-x\|_2^2 + \|\pi_c(y)-y\|_2^2 \leq \|x-y\|^2$, $\forall x \in \mathcal{C}$ In which $\pi_c(y)$ is ...
0
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1answer
38 views

Showing existence of a projective [duplicate]

Let $X$ be a normed space and $Y$ be a finite dimensional subspace of $X$. Show that there is a projective $P\in B(X)$ such that $Im P=Y$. Hint: First Solve for $dimY=1$ then generalize the solution ...
1
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1answer
20 views

For a family of projections how to prove $\vee(I-E_a) \ge I-\wedge E_a$?

In the book "Fundamentals of the Theory of Operator Algebras" by Richard V. Kadison and John R. Ringrose at page 111 (a screenshot is attached below) the authors claims: Since the map $E \...

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