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Questions tagged [projection]

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How to find Orthogonal Projections?

I am very confused regarding the topic orthogonal projections, so I will be really thankful if someone could help me. In my script is written that in order to find the orthogonal projection of a ...
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How to show equivalence between two programs?

Consider the following space $A = \{(x_1,x_2,x_3)\in \mathbb{R}^3|x_1+x_2+x_3 = 1\}.$ Then say that we want to minimize a function $J(y):\mathbb{R}^{3}\to \mathbb{R}$ subjected to the constraint that $...
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Central Projection and Existence of the Vector Space

My definition of a central projection is the following: Let $U_1, U_2 \subseteq \mathbb{P} (V) := \{ U \subseteq V \, \text{ subspace } | \, \text{dim}_K(U) = 1 \}$ be two projective subspaces ...
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Is this a known (valid?) projection?

I've seen in several astrophysical articles the following projection used to transform (spherical) equatorial coordinates $(\alpha,\delta)$ of a dataset, to rectangular $(x,y)$ coordinates: $$ x = (\...
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How to find the projection along the following vector subspace?

I am given with the inner product, $$\phi(a,b) = a_1b_1+a_2b_3 + a_3b_2$$ where $a=(a_1,a_2,a_3)\text{ and } b= (b_1,b_2,b_3)\in \mathbb{R}^{3}.$ Consider the vector space $F = \text{span}(1,1,1).$ ...
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How to project a vector $x$ onto the eigen-space, i.e. $x = \sum_{i=1}^{n} \langle w_i, x \rangle v_i$?

Assume $A$ is a complex-valued square matrix, i.e. $A\in \mathbb{C}^{n\times n}$, and $A$ has a full set of eigenvectors denoted as $V=[v_1, v_2, \cdots, v_n]$. Then we known the following facts \...
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Projecting onto a plane

Given vectors are $v_1 = (1,-2,1,3) $ , $v_2=(0,3,1,-1) $ and $v_3=(1,1,-1,-1) $ in $ R^4$ with the standard inner product. The question is how to find the orthogonal projection of $v_3$ on the ...
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simplest way to compute if a point can project onto a line segment

Given a point P in three-dimensional space, and a line segment from A to B, what is the simplest way to calculate if the projection of P onto AB would lie on the line segment itself (endpoints ...
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Orthogonal projection of the vector $p=(1,0,0,0)$ onto the subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$

I got a subspace $W=[(1,-3,0,1),(1,5,2,3),(0,4,1,1),(1,-2,0,4)]$ and I want to make an orthogonal projection of a vector $p=(1,0,0,0)$ onto $W$ and onto the orhhogonal complement of $W$. ...
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How to get a new matrix with the diagonal elements of another one, and zeros in the rest of the entries? [closed]

I'm NOT interested in how I can do this in MATLAB. I would like to know how, given a matrix $A\in\mathbb{R}^{n\times n}$, I can extract only the elements on its diagonal, put them in a new matrix $B$ (...
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Proving a projection onto subspace if $P^2 = P$

Suppose that V is a vector space, and M is a subspace of V . A transformation $P : V \to V$ is called the projection of V onto M if (i) there exists a subspace N such that every vector v ∈ V can be ...
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(When) is the bicommutant of a linear operator equal to the set of “simpler” operators?

Let $V$ be a $\Bbbk$-linear space and $T\in \mathrm{End}_\Bbbk(V)$. Definition. Say an operator $S\in \mathrm{End}_\Bbbk(V)$ is simpler than $T$ if every $T$-invariant (internal) direct sum ...
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Linear Algebra - Proving a projection given a linear transformation

Suppose that $V$ is a vector space, and $M$ is a subspace of $V$. A transformation $P:V \rightarrow V$ is called the projection of $V$ onto $M$ if (i) there exists a subspace $N$ such that every ...
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Is this property of a set of vectors is independent of the basis choice?

Let $\{v_1,\ldots,v_{n}\}$ be a set of orthonormal vectors in $\mathbb{R}^n$. If these vectors have this property that for some basis $\{a_1,\ldots,a_{n}\}$, for any $j\in \{1,\ldots,n\}$, $$\sum_{i=...
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Finding two projections of a vector that their resultant is the first vector

I have a vector with defined magnitude and direction and want to project it into two vectors. The resultant of those two new vectors should be equal to the first vectors. All of the 3 vectors have the ...
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Convex sets - metric projections

In my course of optimization I got the following definition of metric projections Let $\textbf{C}$ be a closed convex subset of a Hilbert space $\textbf{V}$ . For any $\textbf{x} \in \textbf{V}$, ...
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Motivation for Projection definition in polytopes

In the following Projection, Lifting and Extended Formulation in Integer and Combinatorial Optimization by EGON BALAS polytope projection definitions are given: Given a polyhedron of the form $$Q := \...
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Algorithm to sort 3d segments from back to front depending on z-Order

I have a set of $n$ segments in 3D space with start and end points $A_i$ and $B_i$ $(i \in \{1, ... n\})$ in $\mathbb{R}^3$. Those segments are projected on screen and their $x$ and $y$ coordinates ...
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How to project a vector on a set defined by linear inequality constraints through KKT conditions?

I need to find the projection $x \in \mathbb{R}^{k}$ of a vector $z \in \mathbb{R}^{k}$ on the set defined by $Y \cdot x \geq 0$ where $Y$ is a (given but no specific property) matrix of size $m \cdot ...
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$\cap$-Stable Generator for Canonical Product $\sigma$-Algebra

Let $D$ be a (possibly uncountable) topological space with Borel $\sigma$-algebra $\mathcal{B}(D)$. Let the space of functions from $D$ to $\mathbb{R}$ $$ \mathbb{R}^{D}: = \{f \mid f : D \to \...
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Projecting a point in 3D unto a 2D image with a camera that can move.

Having a camera that can only move on the shell of a sphere, is it possible to calculate where on the 2D image a dot in the 3D space would be? The camera's position is known as the two spherical ...
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Derivation of the relation between the displacement of a 2D point and the 3D counterpart

In this paper, a relation is specified between the movement of a projected point on an image and the movement of the corresponding vertex of a 3D surface: Being $\Pi^{-1}_{i,S}$ the antiprojection ...
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9.5.2 Computing a subset basis in “Coding the Matrix” by Philip N. Klein.

I am reading "Coding the Matrix" by Philip N. Klein. He wrote "A simple induction shows that, for $j=1,\dots,k$, the vector added to $vstarlist$ is $v_{i_j}^{*}$." I cannot prove this fact. ...
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Show $A^TP_{R(A)}=A^T$ for all $A$ in $\mathbb{R}^{m \times n}$?

Prove that right multiplication of the orthogonal projector that maps vectors to the range of $A$ by $A^T$ backs to $A^T$. Algebraically, $$A^TP_{R(A)}=A^T$$ where $R(A)$ is the range of matrix $A$. ...
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2answers
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How to show $P_MP_N=0$ if $M \perp N$?

Let $M$ and $N$ be two subspaces of a vector space $V$, and consider the associated orthogonal projectors $P_M$ and $P_N$. Proof if $M \perp N$ ($M$ is perpendicular to $N$), $P_MP_N=0$? In other ...
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How to show $x \in R(P)$ when if $P$ is orthogonal projector?

For an orthogonal projector $P$, if $\|Px\|_2=\|x\|_2$, show that $x \in R(P)$, where $R(P)$ is the range of $P$, $x \in \mathbb{R}^n$, and $P$ is an $n \times n$ matrix. We need to show there exist $...
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Derivative of a function including Euclidean projection?

Let $C$ be a closed convex set in $\mathbb{R}^n$. Let $z$ be a point in $\mathbb{R}^n$. Definition: Euclidean projection of $z$ onto $C$ is defined as $$ \pi_C(z)=\arg\min_{x\in C} \|z-x\|_2 $$ ...
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Inequality on pairs of projections in Kato's book

I do not understand an argument (p. 58, l.2--3) regarding two "close" projections, in the proof of Theorem I.6.34, pp. 56--58, Kato's book "Perturbation Theory for Linear Operators". The setting is ...
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1answer
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what's spectral axiom

I encounter a proposition in an article: For any $0\leq x\leq 1$ in a C*-algebra enjoying the spectral axiom, there are projections $(e_n)$ such that $$x=\sum_{n=1}^{\infty}\frac{1}{2^n}e_n.$$ ...
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Math notation to define the operator that extract component of a vector?

Say $\mathbf{x} \in \mathbb{R}^n$, what is the common notation to extract the first component as an operation? Something like $\mathcal{P}_j = \mathbf{x}_j$? $\mathbf{x}_j$ is the j-th component. I ...
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1answer
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Computing projection onto the following closed convex set

Let $\mathbf{S}^n$ denote the space of symmetric, real-valued $n \times n$ matrices. Consider the closed convex set $$ \mathcal{C} := \{(X, x) \in \mathbf{S}^n \times \mathbf{R}^n : X \succeq xx^T,...
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Component-Wise 3D Vector Projection

I want to define the projection of a vector $\mathbf{v} \in \mathbb{R}^3$ onto the line $\mathbf{r} \in \mathbb{R}^3$ in terms of the components of $\mathbf{v} = (v_x, v_y, v_z)$. In 2D, this looks ...
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Projection of a circle via Matlab

I am currently stuck at this exercise - I dont know where to begin, which forumals should I use :/ You have to implement the projection of a circle as viewed by a camera in 3D with different angles ...
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1answer
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Best linear prediction as a projection in a Hilbert space $L^2$

Consider two random variables $Y$ and $X$. In the context of the best linear prediction, if we would like to predict $Y$ given $X$ known, we derive the solution solving the following minimize problem ...
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1answer
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How to show $\text{rank}(p)=\text{trace}(p)$ for every projector $p$ defined on $\mathbb{R}^n$? [duplicate]

I know this is an old question and there are several answers for this using eigenvalues and matrix factorization but they have not taught in my matrix analysis course yet. Therefore, my question would ...
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1answer
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Projection map for polynomial rings

Let $K$ be a field and Consider the projection map $\pi_{i,j} : K[X]/(X^i) \to K[X]/(X^j)$, for $j \leq i$. This is well-defined since $(X^i) \subseteq (X^j)$. I'm wondering what it looks like, is it ...
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1answer
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Orthogonal Projection in Hilbert Space to prove weak convergence

Let $H$ be a hilbert space and $(h_{n})_{n\in\mathbb{N}}$ be a bounded sequence in $H$. Define $H_{0}:= \text{cl}(\text{span}(h_{1},h_{2},...))$. Then, $H_{0}$ is a separable space since the set of ...
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Understanding the orthogonal projection vector derivation

As you can see below, $z$ is the projection of $x$ onto $y$... I am trying to derive the orthogonal projection formula based on things I already know. Calculating $cos(\theta)$ is trivial... $$cos(\...
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1answer
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Let $P ⊆ X ×Y$. Does $π_1(P)×π_2(P) = P$? Give a proof or a counterexample.

Proofs and fundamentals, exercise 4.2.5. I need your help, maybe it's false. Let $X$ and $Y$ be sets, let $A ⊆ X$ and $B ⊆ Y$ be subsets and let $π_1 : X × Y → X$ and $π_2 : X × Y → Y$ be projection ...
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Matrix of a non-orthogonal projection and idempotence [closed]

Can find materials with the proof for LA question. So: Is this true that the matrix of a non-orthogonal projection is idempotent?
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Projection map from projective space is not well defined?

Let $f: \mathbb{P^2} \to \mathbb{P^1}$ send $(x,y,z)$ to $(x,z)$. My book is telling me that this map is not well defined at $(0,0,1)$. How is this the case? Here $\mathbb{P}^1$ and $\mathbb{P}^2$ are ...
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1answer
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Is a von Neumann algebra a closed linear span of pairwise orthogonal projections?

It's well known that a von Neumann algebra is a closed linear span of its projections. Can we require these projections to be pairwise orthogonal? that is, can we find a set $\mathscr P$ ...
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Least-Squares Opposite

Is there an opposite formulation of least-squares projection where the distance between each point–say, $(x_{i}, y_{i})$– and the subspace that it's projected onto (e.g. a line through the origin) is ...
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Projection on the GNS subspace

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. If $\omega$ is a positive linear functional on $\mathcal{A}$, then we may perform the so-called GNS construction in order to obtain ha Hilbert space $\...
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1answer
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Discrete Chebyshev inequality as double projection

Let $a_1< a_2<\ldots< a_n$ and $b_1, b_2, \ldots, b_n$ be real numbers. Then \begin{align*} &b_1< b_2< \ldots< b_n\Rightarrow \frac{a_1b_1 + a_2b_2 +\ldots + a_nb_n}{n}> \...
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Finding the matrix for a projection map, $T^2=T$.

Doing some revision, and was wondering if someone could help me work through this question. You are given that $E:V\longrightarrow V$ is a projection map, so that $E^2=E$. You are then asked to ...
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1answer
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Projection map is closed or not?

If we consider projection of $\Bbb R\times \Bbb R$ onto the $x$-axis(codomain is $\Bbb R\times \Bbb R$) the map is not closed as we know the image of a hyperbola ($xy=1$, closed) is not closed. In my ...
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Predicting a non-causal stationary autoregressive process

Consider the unique stationary solution to the following relation $$X_t = \theta X_{t-1} + Z_t$$ where $(Z_t)_{t \in \mathbb{Z}}$ is a white noise series and $\lvert \theta \rvert > 1$. I am trying ...
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Reference Request: Projection from $L^1$ onto $L^2$

Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space. Since $L^2(\Omega,\mathcal{F},\mathbb{P})$ is a subspace of $L^1(\Omega,\mathcal{F},\mathbb{P})$, is there a well-...
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1answer
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Finding the Orthogonal Projection from a Matrix where $ a_{ij} = \vec v_i\cdot \vec v_j$

I am working on a problem that is asking to find $\operatorname{proj}_Vv_3$ where the matrix $$A=\begin{bmatrix} 3&5&11\\ 5&9&20\\ 11&20&49\end{bmatrix} $$ has entries $a_{ij}=\...