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Questions tagged [projection]

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Kernel of orthogonal projection on an eigenspace

let $Q$ be a $d\times d $-matrix and $P:\mathbb{R}^d \to \mathbb{R}^d$ be the orthogonal projection on the eigenspace $E_0 $ of $Q$. Why is the kernel of the projection the sum of the other ...
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1answer
17 views

Norm of the sum of orthogonal projections

Let $H_\lambda$ be a closed subspace of a $\mathbb R$-Hilbert space $H$ for $\lambda\ge0$ and assume that $(H_\lambda)_{\lambda\ge0}$ is nondecreasing and right-continuous, i.e. $$\bigcap_{\mu>\...
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0answers
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Projecting a selection of points from a regular 2D grid onto a line

I would like to: start with a regular 2D grid like shown in the picture chose a line of slope $\tan \alpha$, and project, and a window of acceptance (grey area) project the points of the grid within ...
4
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1answer
64 views

Does the closest point on a subset change continuously?

$\newcommand{\til}{\tilde}$ Let $(X,d)$ be a metric space, and let $S \subseteq X$. Suppose that every point in $X$ has a unique closest point in $S$, which we denote by $\tilde s(p)$. Is it true ...
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0answers
44 views

Covariance matrix and projection

I have troubles understanding a geometrical meaning of a covariance matrix. Let's say we have a data set containing two points (-1,1), (-1,2) and write them in to the matrix $$D = \begin{bmatrix} -...
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2answers
25 views

$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$ if $H$ is Hilbert and $P,Q$ orthogonal projections.

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ orthogonal projections. Show that : $$PQ = 0 \Leftrightarrow P(H) \bot Q(H)$$ Attempt-Thoughts : $(\Rightarrow)$ Let $PQ = 0$. ...
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1answer
34 views

General equation for projection of regular grid onto a line?

I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $\tan \alpha$: I would like to derive a mathematical ...
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2answers
34 views

Defining a kind of “projection of a measure” in a precise way

Suppose we have a probability measure $$\mu: \mathcal{S} \times \mathbb{R}\rightarrow [0,1],$$where $\mathcal{S}$ is some countable set. In some personal, handwritten lecture notes I'm reading it is ...
2
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1answer
20 views

$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$

Exercise : Let $H$ be a Hilbert space and $P,Q \in \mathcal{L}(H)$ are orthogonal projections, then show that : $$PQ \; \text{orthogonal projection} \; \Leftrightarrow PQ = QP$$ Seeking a formal ...
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Independent projections

Suppose, we have a matrix $X\ (n\ \times\ n) $. There are r < n independent columns(therefore r is rank).And we project our vectors with $P\ (n\ \times\ n) $ operator = $I - E$ where E is the just ...
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1answer
30 views

Let $A$ be an operator, such that $A^{\dagger}A$ is a projector. Show that $AA^{\dagger}$ is also a projector

There is a hint to this problem which I don't know how to interpret. The hint is $$ \text{Hint: Show that} \quad A | \phi \rangle = 0 \leftrightarrow A^{\dagger}A=0. $$ Attempted solution(without ...
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Object's plane on image [closed]

I have a segmented image. So, for example, I know 2D coordinates of pixels for road on image. Now I want to know a plane of this road. How can I do this? I assume that decision rests on Ransac, The ...
2
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1answer
28 views

Free probability and projections

Let $(\mathcal{A},\varphi)$ be a free probability space, where $\mathcal{A}$ is a von Neumann algebra and $\varphi$ a finite and faithful trace. Let furthermore $p\in\mathcal{A}$ be a projection. ...
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1answer
24 views

Maximize inner product subject to constraint

Let $a\in\mathbb{R}^{d}$ and $K$ be an arbitrary subset of $\mathbb{R}^{d}$. My question is related to the following optimization problem: \begin{equation} \max_{x\in\mathbb{R}^{d}}~a^{\top}x\quad \...
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Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
0
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1answer
21 views

How to prove $||P_\Omega(v)-P_\Omega(u)|| \le ||v-u||, \forall u,v \in \mathbb{R}^n$?

Suppose $\Omega$ is a closed convex set in $\mathbb{R}^n$. Let $P_\Omega(u)$ represent the projection of $u$ onto $\Omega, \forall u \in \mathbb{R}^n$. That is to say $P_\Omega(u)= \underset{v \in\...
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Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$}

Consider the surface $S$ that is the intersection of $x^2 + y^2 + z^2 = 4$ with the cylinder $(x-1)^2+y^2 \leq 1$ Find the Area of the orthogonal projection of $S$ on $H$ = {$3x + 2y + 100z = 0$} I ...
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1answer
48 views

Orthogonal decomposition with a special inner product

Assume that we are working on $\mathbb{R}^p$ with an inner product induced by the positive definite matrix $\mathbf{G}$, i.e. for $\mathbf{f}, \mathbf{g} \in \mathbb{R}^p$ we define $\langle \mathbf{f}...
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1answer
41 views

Orthogonal projection of a vector on a linear subspace

I know this question has already been asked before but I'm not quite sure I've understood them well enough. I want to find the projection $\vec{v}_{proj}$ of $\vec{v}\in V$ on the span of the vectors ...
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Projected gradient descent with matrices

I have a scalar function $f(\rho) = Tr(\rho H) + c\ Tr(\rho\log\rho)$, where $Tr$ is trace, $\rho$ is a positive semidefinite matrix with trace 1 and $H$ is a Hermitian matrix and $c$ is a postive ...
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1answer
30 views

Prove projection of convex hull = convex hull of projection

I'm not sure how to show this: $proj_x(conv(S)) = conv(proj_x(S))$ where S $\in R^{n+p}$
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Inverse Perspective Mapping mathematics

I would like to know the mathematical formulation of inverse perspective mapping (IPM). in [1] (equation 1 and 2) why "n-1" is used as image resolution instead of "n"?? and the second question is ...
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2answers
25 views

Hilbert Space: Orthogonal projection is linear

Let $X$ be a Hilbert space and $A\subset X$ a closed subspace show that the orthognal projection $P:X\rightarrow A$ is linear. Now I know that $x-P(x)\in A^{\perp}$. The lecture notes go on by saying ...
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2answers
15 views

Show that $A$ is a difference between two orthogonal projections.

Let $V$ be a finitedimensional complex vector space. Linear operator $A \in L(V) $ is hermitian and unitary. Show that $A$ is a difference between two orthogonal projections. The questions seems ...
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0answers
47 views

projection from a point to a constrained hyperplane

I am trying to find the closest point on the following constrained hyperplane to a general point $\vec x$ : $$ \vec \omega \!\cdot\! \vec 1 = 1 \ \ s.t \ \ \alpha_i \le\omega_i\leq\beta_i $$ $$ 0\...
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1answer
36 views

Integrability with respect to a spectral measure

Let $H$ be a $\mathbb R$-Hilbert space. If $(\mathcal D(A),A_i)$ is a symmetric linear operator on $H$, write $A_1\le A_2$ if $$\langle A_1x,x\rangle_H\le\langle A_2x,x\rangle_H\;\;\;\text{for all }x\...
2
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1answer
52 views

If $(H_λ)_{λ≥0}$ is a spectral decomposition and $π_λ$ is the orthogonal projection onto $H_λ$, then $t↦π_λ$ is increasing and right-continuous

Let $H$ be a $\mathbb R$-Hilbert space. If $(\mathcal D(A),A_i)$ is a symmetric linear operator on $H$, write $A_1\le A_2$ if $$\langle A_1x,x\rangle_H\le\langle A_2x,x\rangle_H\;\;\;\text{for all }x\...
4
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2answers
61 views

Find the unit vector within a subspace with the minimum norm projection onto another subspace

Let $W$ and $V$ be subspaces of $\mathbb{R}^n$ with dimensions $m$ and $p$ respectively. I want to find the unit vector in $W$ whose projection onto $V$ has the minimum Euclidean norm. From geometric ...
0
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1answer
39 views

Is every orthogonal projection continuous?

Is there an example of a non-continuous linear operator $\pi$ on a $\mathbb R$-Hilbert space $H$ with $\pi^2=\pi$ and orthogonal null space and range? Clearly, if the range is closed, then $\pi$ is ...
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13 views

Compilation of known facts about projectors

I would like to make a compilation of useful properties about projectors (either orthogonal one and non-orthogonal one). Let $P \in M_{n,p}(\mathbb{R})$, $P^2 = P$. Here are the properties I know :...
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Covering from Dense Projection

Let $V$ be a Banach space, $W$ a (strict) subspace of $V$, and $U$ a dense proper subset of $V$. When is does there exist a (linear) projection $P^V_W:V\twoheadrightarrow W$, such that $$ P_W^V(U)=W ....
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37 views

What could be the formal mathematical meaning of “onto over?”

I found an author who likes to use the phrase "$\underline{\textbf{onto over}}$" For example, the textbook author might write: The set $X$ projected $\underline{\textbf{onto over}}$." the variable ...
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2answers
95 views

Viewing a circle from different angles - is the result always an ellipse?

Take a piece of rigid cardboard. Draw a perfect circle on it. Hold it up, and take a picture, with the cardboard held perpendicular to the direction we're looking. You get a photo that looks like ...
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2answers
45 views

Inequality with projections in Hilbert Space

Problem. Let $X$ be a Hilbert space and $\emptyset \neq K \subseteq X$ be closed and convex. Then, $$ \|P_Kx - P_Ky \| \leq \|x-y \|$$ for all $x,y \in X$. Here, $P_K$ is the projection from $X$ onto $...
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1answer
40 views

projection operator a and b on vector space V s.t. ab = -ba, prove that ab = ba = 0 [closed]

Assume a and b are projection operators on a vector space V, such that ab = -ba. prove that ab = ba = 0
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Riemann hyper-spheres for hyper-complex numbers?

Today I learned that the Riemann sphere can map the extended complex plane to the surface of a sphere. It's straight-forward to show that an analogous mapping can be found for a line and a circle. I'...
0
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1answer
19 views

linear algebra - orthogonal projection

Sorry if I wrong in the English terms.. I have V which is Internal product space. W = sp{(1,0,-1),(1,1,0)} is a base, subspaces of V. v = (-3,0,5) is in V. I need to find the orthogonal projection ...
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2answers
32 views

Ellipse projection knowing semiaxes vectors

I'd like to know the semi-projection of a tilted ellipse on $x$ and $y$ axes, called as $O_\parallel$ and $O_\perp$, knowing the vectors $\vec{\zeta}=(\zeta_x,\zeta_y)$ and $\vec{ \nu}=(\nu_x,\nu_y)$ ...
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1answer
23 views

Metric projection from space of bounded functions to finite-dimensional linear space

Apologies if the answer is obvious or should be easy to find, but so far I've had no luck. Let $X$ be a subspace of $\mathbb{R^k}$ for a finite $k$ and let $\mathcal{B}(X)$ be the Banach space of ...
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3answers
58 views

Give the standard matrix of the projection $T:\Bbb R^3 \to \Bbb R^3$ that projects a vector on the plane $x+y+z=0$

I'm trying to figure this one out guys with no luck. Give the standard matrix of the projection $$ T:\Bbb R^3 \to \Bbb R^3 $$ that projects a vector on the plane $x+y+z=0.$ I tried to make a basis ...
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1answer
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Properly infinite projection/von Neumann algebra

The definitions I am using are: Def: a projection $e \in M$, von Neumann algebra, is said to be purely infinite, if given any projection $p \in Z(M) \subset M$, then $pe$ is finite if and only if $pe=...
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1answer
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Is there a special name for the orthogonal projection matrix onto the unit vector?

Many problems in multivariate analysis involve the $n \times n$ matrix: $$\mathbf{M} \equiv \boldsymbol{I}_n - \frac{1}{n} \mathbf{1}_{n \times n}.$$ This is an orthogonal projection matrix onto the ...
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0answers
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Projection on the hyperplane H: $∑x_i=k$ [duplicate]

Consider the hyperplane $H$: $∑x_{i}=k$, where k is a constant and $i=1...n$. How do we project a vector onto this subspace? More precisely, how do we compute the projection matrix $P$?
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1answer
56 views

Boundary of the projection of a body

So I have a body defined in a coordinate system and I project this body to the XY plane. How do I find the boundary of the projected shape? Which theory is this topic related to? Is there any paper/...
4
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1answer
76 views

What exactly is a non-linear orthogonal projection?

In a Hilbert space of bounded integrable functions, let $P$ be an operator such that $$P(f(x)) = \frac{f(x)+|f(x)|}{2}$$ The complement of $P$ can be written as $Q = I - P$, hence $$Q(f(x)) = \frac{...
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1answer
51 views

Norm estimate for a product of two orthogonal projectors

Let $H$ denote a Hilbert space. Consider two orthogonal projectors $\,P,Q\in\mathscr L(H)\,$ such that $H=\operatorname{Im}P\oplus\operatorname{Im}Q\,,$ that is both $\,\operatorname{Im}Q\,$ and $\,\...
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2answers
53 views

Show that $f$ is an orthogonal projection

$A=M(f,B)=\frac{1}{3}\begin{pmatrix} 1 & -1 & 1\\ -1& 1 & -1\\ 1& -1 & 1 \end{pmatrix}$ Show that $f$ is an orthogonal projection on a line to be determined. How to solve ?...
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1answer
50 views

A “Crookedness criterion” for a pair of orthogonal projectors?

If $P$ is an orthogonal projector on a Hilbert space $H$, then $\,\operatorname{im}P=(\ker P)^\perp\subset H\,$ is a closed subspace, also called the support of $P$. And vice versa: Every closed ...
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1answer
27 views

Project a point onto a plane, given some constraints

I tried some of the formulas answered here, but none of them works within the constraints of my problem. Here is it: I got two points - P1 and P2 and a line - [Pa, Pb]. Inbetween these points, I ...
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1answer
25 views

Prove that linear operator T is the projection Operator

I got question prove or disprove Let be $T:\mathbb R^n \to \mathbb R^n$ linear oprator and let be $U\subset\mathbb R^n$ hyperplane. If known that $Image(T)=U$ and also that for every $u\in U$, $...