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Questions tagged [projection]

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Orthogonal projection of an ellipsoid

Suppose an ellipsoid given by $\{{(x,y,z)| {x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}}}\}$, find the area of the orthogonal projection of the ellipsoid on the plane ${2x+4y-5z=10}$. What is the right ...
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Finding $L_1$ projection (i.e. $\arg\min_\mathbf{x} ||\mathbf{Ax} - \mathbf{b}||_1$)

Is there an efficient way or algorithm to find the $L_1$-projection from a vector $\mathbf{b}$ with a transformation matrix $\mathbf{A}$? In other words, how to find a vector $\mathbf{x}$ that ...
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How to find the equation of an image under a central projection

Let $\pi:\mathbb{P}^3 \to V(x_2) \cong \mathbb{P}^2$ the linear projection with center $P =(0:1:0:0)$. Find the equation for the image of $C=\{(s^3:s^2t:st^2:t^3)|~(s:t) \in \mathbb{P}^1 \}$ under $\...
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Orthogonal projection in Hilbert space

Projector $P\neq0$ is orthogonal projection if and only if satisfied some of the following mutually equivalent conditions: a) $P$ is self-adjoint, $P=P^*$ b) $P$ is normal, i.e. $P^*P=PP^*$ c) $P$ ...
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1answer
28 views

Deformation Retraction and Projection/Closest Vector

How do I compute the projection/closest vector to a subset? I have been thinking about this for far too long without any progress. If it helps, I am working in $\Bbb{R}^2$, but I would like formalue ...
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I have 5 dimension points (x,y,z,w,v)

I have an image of made of points of (x,y). I want to combine the hsv color space of these points so that I have (x,y,h,s,v) Now how do I project these 5D points onto a 2D image? I found some link ...
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simple hypergraph projection [closed]

I have been struggling quite a while with a question, which I suspect has a simple answer to: I have a Graph G = (X,E,Ψ) with E being a family of subsets of X and Ψ being a mapping between two ...
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Projections, Prove $P$=$P^2$ [duplicate]

$f$:$C^n$ → $C^n$, $f$($x$) := $x_u$.Let $P$ ∈ ${C}^{n×n}$ be such that $f$ = $f_p$ . Show that $P^2$ = $P$ Does someone have an idea about this Problem here? I thought that $P$= $A$ $A^t$ but, I am ...
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1answer
25 views

Matrix associated with a projection mapping

From S.L Linear Algebra: Find the matrix associated with the following linear maps. The vectors are written horizontally with a transpose sign for typographical reasons. (a) $F:\mathbb{R}^4 \...
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1answer
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Forward and Backward Projections

I have the transform functions (forward and backward projections) such as: $$FP\{f(x,y)\} = \int_{-\infty}^{\infty}f(r\cos(\theta) - z\sin(\theta), r\sin(\theta) + z\cos(\theta))dz$$ $$BP\{g_{\theta}(...
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1answer
28 views

Projection Formula

Does someone know if in the problem the projection of x onto U is defined like that : $x_u = \displaystyle \frac{\langle x,u\rangle}{u. u}$ $u$ Problem: Let $U,V\subset\mathbb C^n$ be two ...
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1answer
39 views

General Projections

I have the following problem, however I cannot understand what I exactly have to show, as I am not sure what $x_u$ means. So can someone tell me if it is the projection of $x$ onto $U$, or is it ...
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Random projection of a fixed point

In the book "High-dimensional probability by Vershynin", page 111, in the proof of Johnson-Lindenstrauss Lemma, let $E$ be a random $m$-dimensional subspace in $\mathbb{R}^n$ uniformly distributed in ...
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Least square problem constrained to projection matrices

Some times in engineering, it is important to find an optimum subspace in which projecting on it satisfies some properties. Let known matrices $A$ and $B$ belong to $\mathbb{R}^{p\times n}$ and $\|\...
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1answer
29 views

Linear Algebra Eigenvalues from a geometric description

Goodmorning, I have a question related to Linear Algebra: The line L through the origin and (1,2) is given. Furthermore we know that for any vector v the vector Av is given by the projection of v ...
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how to find the orthogonal projection of u onto v

I would love some help with a question I dont know how to answer. Let $ u=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}^T,V= \begin{bmatrix} 1 & i \\ -i & 1 \\ 1 & 0 ...
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Projection associated to the decomposition $H=M⊕N$

Let $H$ be a Hilbert space and let $M$ and $N$ be two closed subspaces in $H$ such that $H=M⊕N$. I'm trying to find a formula giving $P_{M,N}$ (the projection onto $M$ with respect to $N$) in terms ...
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2answers
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orthogonal projection of a vector onto a plane

I am trying to find the orthogonal projection of a vector $\vec u= (1,-1,2)$ onto a plane which has three points $\vec a=(1,0,0)$ ,$\vec b=(1,1,1)$, and $\vec c=(0,0,1)$. I started by projecting $\vec ...
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If $\overline{M}=A\oplus B$ what can be said about $M$'s decomposability?

Let $X$ be a Banach space, and $M$ a subspace in $X$ such that $\overline{M}% =A\oplus B$, where '$\oplus $' designates the topological direct sum. Do we have $M=A_{0}\oplus B_{0}$ such that $\...
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Projection in $\ell_1$ norm onto linear subspace of $\mathbb{R}^n$.

In the context of $\mathbb{R}^n$, fix a linear subspace $\mathcal{U}$ or arbitrary dimension $r \leq n$ and some vector $\mathbf{x} \in \mathbb{R}^n$. What is the most efficient way to compute the ...
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1answer
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Projection of a $L^2$ function over a set of constant functions

I don't know how to solve the following exercise: Let $X$ be the subspace of $L^{2}(0,1)$ of a.e. constant functions. What is the projection over $X$ of a function $u \in L^2(0,1)$? First of all, $...
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What is the formula for projection onto spectraplex?

A spectraplex (special case of spectrahedron) is the set of all positive semi-definite matrices whose trace is equal to one. Formally, let $$ S=\{\textbf{W} \in \mathbb{R}^{d \times d} \mid \textbf{W} ...
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Frobenius norm and singular values

I study about random projection and i m really confuse about the relationship between Frobenius norm and singular values. The book say that the $||M||_f^2 $ and $\sigma$ had a correlation. I found ...
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1answer
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On proposition I.1.2 of “Quantum Groups” by Christian Kassel

I am working through Christian Kassel's textbook on Quantum Groups. The Proposition states that 5 statemens are equivalent. The two I am having trouble with are as follows. 1.For any pair $V'\subset ...
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2answers
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Inequality involving projectors on a Hilbert space isomorphic to $\mathbb{C}^n$

Suppose we have Hermitian operators $P_1$ and $P_2$ on Hilbert space $\mathbb{H} \cong \mathbb{C}^n$, such that: $P_1^2=P_1$, $P_2^2=P_2$, $\{P_1, P_2\} = 0$. Let $x$ be a unit $L_2$-norm vector. ...
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Volumes of bodies in $\mathbb R^3$ and $\mathbb R^6$

In $\mathbb R^3$ define a brick to be of the form $[a,b] \times [c,d] \times [e,f] $ Define a body to be a finite union of bricks. For any object $X$, let $|X|$ denote its "volume". Further, denote ...
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Finding the length of the projection of the vector onto a line using parametric equation

What is the length of the projection of the vector $(3, 4,-4)$ onto a line whose parametric equation is the following? $$\begin{aligned} x &= 2t + 1\\ y &= -t + 3\\ z &= t - 1\end{...
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1answer
51 views

How to show $x_* \in R(A^T) +z $ is the global minimizer of $\min \frac{1}{2}\|x-z\|_2^2$ over $Ax=b$?

Consider the following problem: $$ \min_{Ax=b} \frac{1}{2}\|x-z\|_2^2 $$ where $z \in \mathbb{R}^{n}$ is a given vector, $x \in \mathbb{R}^{n}$, $A \in \mathbb{R}^{m \times n}$ and $b \in \mathbb{R}^{...
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Random Projection algorithm is strictly not a projection?

Current implementations of the Random Projection algorithm reduce the dimensionality of data samples by mapping them from $\mathbb R^d$ to $\mathbb R^k$ using a $d\times k$ projection matrix $R$ whose ...
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1answer
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Norm limit of sequence of orthogonal projections on Hilbert space “contractive”

Let $(P_n)_{n \in \mathbb N} \subseteq B(\cal H)$ be a sequence of (orthogonal) projections on a (separable) Hilbert space such that $\left\Vert P_{n}\xi\right\Vert \rightarrow C\left\Vert \xi\right\...
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Projection on convex sets with equality and inequality constraints

I want to find the projection of a vector called "a" on a closed and convex set with linear constraints. The set is in the following form: \begin{array}{ll} & Ax = b \\ & Bx \le d \\ &x \...
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Show that any projector onto subspace L is parallel to M

Assume that $\mathbb{R}^n$ is represented as the \emph{direct} (but not necessarily \emph{orthogonal}) sum $M_1 \dotplus M_2$ of two its subspaces $M_1$ and $M_2$. In particular, every $x \in \mathbb{...
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1answer
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Unicity of the projection on a closed convex subset of $L^p$

The original problem (not full) statement is the following : Let $(\Omega, \mathfrak{B},\mu)$ a measured space and $p>1$ and let $C$ be a closed convex subset of $L^p$. For $u\in L^p$ let $...
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1answer
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Preimage of a set in the product topology

Claim: When $X ×Y$ is endowed with the product topology $T_{X×Y}$ , the projection maps $p_X : X × Y → X , p_X(x, y) = x$, and $p_Y : X × Y → Y , p_Y (x, y) = y$ , are continuous. Proof: Indeed for ...
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1answer
27 views

Find a matrix of oblique projector

How I should find this oblique projector matrix in the following task? Information at Wikipedia seems to be a little bit complicated and I haven`t found any practical examples for oblique projections. ...
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Prove statements with oblique projections

I can not really understand how to prove the following statements in the task. I have found some information about oblique projections at the Wikipedia, but it is really complicated there and without ...
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Why is the Frank-Wolfe algorithm projection-free while gradient descent isn't?

While reading this article about the Frank-Wolfe algorithm, I did not understand why the Frank-Wolfe algorithm is projection-free, while the gradient descent is not. I think the problem is, that I do ...
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3answers
61 views

Find the orthogonal projection of a function on the set of $L^2$ functions whose integrals are $0$

Given $u\in L^2(0, 1)$, find its orthogonal projection on $V =\{v∈L^2(0,1):\int_0^1 v(x)\ dx=0\}$. For Hilbert spaces it holds a theorem about projections on a closed convex set which states that ...
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Find eigenvalues, kernel and Image of an Orthogonal projection

Let $V$ be a Vector space with inner product and $U$ a subspace. Let $P$ be the orthogonal projection over $U$. Find eigenvalues, kernel and Image of $P$. I know I have to consider the special cases ...
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1answer
53 views

Show distance to the half-space is equal to distance to the set

Let $C \subseteq \mathbb{R}^n$ be a closed convex set, and $x^* \in C^c$ (not in $C$ and its closure). Define the Euclidean distance from $x^*$ to $C$ as $d_C(x^*):=\min_{z \in C}\|z -x^*\|_2$. ...
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Show that $x^*$ is minimizer of the function over $C$ if $x^*=\prod_C (x^*- \gamma \nabla f(x^*))$.

Assume $f: \mathbb{R}^n \rightarrow\mathbb{R}$ be a differentiable convex function and $C \subseteq\mathbb{R}^n$ be a closed convex set, and $\gamma >0$. Consider the following problem $$ \min_{x \...
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1answer
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Some queries related to proof of -“Every Regular space X with a countable basis is metrizable”

We shall prove that X is metrizable by imbedding X into a metrizable space Y;that is by showing that X is homeomorphic to some subspace of Y. Step 1: We prove the following:- There exists a ...
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1answer
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Do we need the Axiom of Choice to guarantee surjectiveness of projections?

Given a collection $\{X_{\alpha}\}_{\alpha\in\Omega}$ of non-empty sets, do we need the Axiom of Choice to ensure that the projections $$\pi_{\gamma}:\prod_{\alpha}X_{\alpha}\longrightarrow X_{\gamma}$...
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When is the nullspace unique in this case?

I am given the following: $$ I - AA^T $$ is a projection matrix onto the orthogonal complement of $< A >$. So the nullspace of $I-AA^T$ is the subspace spanned by the set of vectors $x$ such ...
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Transformation matrix between a 2D and a 3D coordinate system

Supposing I have this data : 2D points (known coordinates): P2D in a 2D coordinate system CS2D. and their corresponding (their ...
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1answer
40 views

Is the representation of a linear affine space unique?

Suppose $S \subset \mathbb R^n$ is a linear affine subspace. I know picking any $s \in S$, $S- s =: U$ is a subspace and we can write $S = s + U$. Now consider writing $s = s_{U} + s_{U^{\perp}}$ ...
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1answer
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Planes, lines, and perpendiculars!

I'm having trouble with this problem - Let $Q = (-2, 3, 4)$, and let $P$ be the foot of the perpendicular from $Q$ to the plane through points $A = (0,1,1), B = (1,1,0)$ and $C = (1,0,3)$. Then $\...
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1answer
29 views

Centralizer of projections

Let $H$ be a Hilbert space and $p, q$ self-adjoint projectors in $B(H)$, i.e. $$p^2=p=p^* \space \text{ and } \space q^2=q=q^*.$$ Suppose they have the same centralizers $C(p)=C(q)$. Is it true that ...
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Projection of matrix in a vector direction

I have a question on matrix projection. For any matrix : $S$ $$S = \sum_i \lambda_i v_i v_i^T$$ So to have a component of this matrix in arbitrary $u_r$ and $u_o$ directions. Here $\{u_r, u_o\}$ ...
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2answers
55 views

Are planes one-sided or two-sided?

I'm looking for clarity here - because, although it seems that planes are treated normally as one-sided, as I understand it, when I read of 'projection onto a plane', it seems that the projection is ...