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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Existence of projection
Suppose vector space $V$ over field $K$ has finite dimension. $W$ and $Z$ are subspaces of $V$ such that $\dim W \geq \dim Z$.
Show that there exist projection $p$ from $V$ to $V$ such that $p(W)=Z$.
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Cauchy-Schwarz inequality with non-standard dot product
I have the following problem:
Let's have the scalar product in space $ℝ^2$ given by the expression:
$$ <x, y> = 2x_1 y_1 + x_2 y_2 + x_1 y_2 + x_2 y_1 $$
For a defined dot product, formulate ...
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How to find an orthogonal transformation between two matrices [closed]
I wonder if there is a way to solve this problem:$$\arg \min_{\alpha, U} \|A - \alpha UB\|_\mathcal F^2 \\ \text{s.t. } \alpha \in \mathbb R, U \in \mathbb R^{n \times n} \text{ and } U^\top U = I$$
...
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Simple statement in the elementary proof of the Johnson-Lindenstrauss lemma (random projections)
In the simple proof of the johnson lindenstrauss lemma written by Sanjoy Dasgupta, Anupam Gupta that can be found here they state the following (p.$62$):
Repeating this projection $O(n)$ times can ...
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Is Frobenius-norm projection Lipschitz continuous under operator norm?
Definition
Let $d \in \mathbb{N}$. Let $A \in \mathbb{R}^{d\times d}$ be a PSD matrix. Define the following operator: $T:\mathbb{R}_{+} \times \mathbb{R}^{d\times d} \to \mathbb{R}^{d\times d} $: Let ...
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Projecting unit $\ell_p$ vector onto $\ell_2$ and measuring distortion in $\ell_p$
Let $p \ge 1$ and let $x \in \mathbb{R}^d$ be a unit vector in the $\ell_p$ norm: $\sum_{i=1}^d |x_i|^p = 1$. Let $v \in \mathbb{R}^d$ be a unit vector in $\ell_2$ in the same dimension: $\sum_{i=1}^...
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Measurable Projection Theorem and the Debut Theorem
The measurable projection theorem (see the George Lowther blog) asserts the following.
Theorem. If $(\Omega,\mathcal{F},\mathbb{P})$ is a complete probability space and $A\in\mathcal{B}(\mathbb{R})\...
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Projection of a Pentagonally-tiled Sphere
I know that a regular pentagonal tiling does not work in Euclidian space, but does work on a sphere. But this got me wondering something that I hope people can help with here, because I can't find any ...
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Perspective image of a cuboid
You are given a perspective image $A'B'C'D'E'F'G'H'$ of a cuboid (rectangular prism) $ABCDEFGH$. The image is such that in the image, in any face, opposite sides are not parallel to each other. In ...
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Orthogonal projection of a conic onto a plane
It is well known that conic sections are the intersections of a plane with a cone. Let the cone be $z^2 = x^2 + y^2$ and the plane be $z = ex + b$. Project orthogonaly the resultant conic onto the $...
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What is meant by projection of this basket? What will the breadth of the basket containing fruits?
It is mentioned that the diameter of the basket is 13" and the projection is 10"
I am clear with the diameter can someone please explain what is mentioned by the projection here?
Also, what ...
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Orthographic projection of a rectangle
Rectangle $ABCD$ is projected using orthographic projection onto a plane that makes a known angle with the plane of the rectangle, as shown in the figure above. Can the lengths of the sides of the ...
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Simple definition of vector rejection: $\vec p + \vec r = \vec v$?
Wikipedia defines vector rejection in a roundabout way. This math.SE answer fills in the rigor. But I'd like to confirm what to me is a very simple definition of vector rejection:
Let $\vec p$ be ...
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Calculating a vantage point from which one shape looks like another
This is my first math stackexchange question, so if it is a duplicate, I will remove it. My question is as follows:
Say I have a point at the origin, in 3D space. Now from this origin point, I have 3 ...
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Spectral Representation of $T$
the spectral family of the operator $T:\ell^2\rightarrow\ell^2$ defined by
$$T(\xi_1,\xi_2,\xi_3,....)=(\xi_1/1,\xi_2/2,\xi_3/3,....)$$
I am trying to get the application of Spectral theorem of ...
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Point on a line closest to the origin, general formula using linear algebra only
Given a line $\ell$ in $\mathbb R^2$ containing points $p, q$, find point $r$ on $\ell$ closest to the origin, using linear algebra only (no calculus).
My answer is below, but I seem to have made a ...
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What is the spectral family of $ \left[ \begin{array}{cc} 0&1&0\\ 1&0&0\\ 0&0&1 \end{array} \right] $?
Spectral Family:A real spectral family (or real decomposition of unity) is a one parameter family $\mathcal E=(E_{\lambda})_{\lambda\in \mathbb R}$ of projections $E_{\lambda}$ defined on a hilbert ...
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In perspective, circle becomes an ellipse. but why isn't the center of circle, center of ellipse? [closed]
In perspective view (a projection), when you look at the circle from different angles, it becomes an ellipse. (also in the world). Most things make sense with perspective, but I don't see why the ...
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Why Do we call a projection of one vector onto another a "component"?
I realize the component of one vector $\vec{a}$ in the direction of the other vector $\vec{b}$ is essentially a projection of $\vec{a}$ onto $\vec{b}$ and I understand how it is calculated. What I don'...
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How could one convert 3d coordinates which are on a plane to coordinates relative to said plane?
I have a plane, defined by ax1+bx2+cx3=d, and a point which I know is on said plane. How could I convert the coordinates of the point to coordinates relative to the plane? I have attempted to find a ...
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how can the scalar projection differ from the absolute value?
A ball travels with velocity given by [2, 1] with wind blowing in the direction given by [3, -4] with respect to some co-ordinate axes. What is the size of the velocity of the ball in the direction of ...
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Conditions, so the operator in L^2 is a projection
Given that $r(x) $is nonnegative smooth function satisfying $r(x)+r(-x)=1$ with support $(-1,\infty)$ and$$(Pf)(x)=r(x)f(x)+t(x)f(-x)$$ from $L^2$ to $L^2$ is a projection , i.e. $P$ is idempotemt and ...
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How $E_\lambda$ is a projection? [closed]
Since the sum of two projections need not be a projection then how at Page 494 it is claimed that $E_\lambda$ is a projection?
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What are the complications with infinite dimensional Hilbert spaces which are not suitable for $T=\sum_{j=1}^n\lambda_jP_j$?
$T=\sum_{j=1}^n\lambda_jP_j$ would not be suitable for immediate generalization to infinite dimensional Hilbert Space s $\mathcal H$ Since in that case spectra of bounded self adjoint linear ...
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A projection operator is linear iff $X$ is a Hilbert space
This question comes from Linear and Nonlinear Functional Analysis with Applications (Philippe G. Ciarlet), Chapter 4, Problem 4.3-4.
4.3-4 Let $\mathcal{P}_n[0,1]=\left\{\left.p\right|_{[0,1]} ; p \...
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How do I prove the inequality?
Say that $y:=$ argmin$_{x\in C} \frac{1}{2}||x||^2_2$.
$C$ is a convex subset of $\mathbb{R}^n$, in Euclidean n-space.
How would I go about to show the following statement for any $x\in C$?
$$||x-y||^...
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How do I find the equation of an ellipse formed by a projection of a tilted rotating circle?
Suppose I have a circle with radius $R$ on $xy$ plane with it's center at $(0,0,0)$. Then I rotate this circle about the y-axis by $0 \le \varphi \le {\pi \over 2}$, and then start rotating it about z-...
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Projection of a function $f$ in a Hilbert space $\mathcal{H}$ on a subspace $\mathcal{D}\subset\mathcal{H}$
I am reading a paper on functional data analysis and they use a Hilbert space as living space for their functions. Then they defined a "dictionary" $\mathcal{D}\subset\mathcal{H}$ where the ...
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Hyperplane Separation Theorem for non-axis aligned rectangles?
I'm trying to understand how the Hyperplane Separation Theorem works.
I found a function on MATLAB that implements it, accepting the parameters
rect1 = [0 0; -1 3; 6 4; 7 1];
rect2 = [12 0; 14 1; 13 ...
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AF algebras have an approximate unity consisting of projections
Let $\mathfrak{A}$ be an AF $C^*$ algebra. Show that $\mathfrak{A}$ has an approximate identity $E_n$ (countable since this is an AF algebra) consisting of an increasing sequence of projections.
My ...
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Proof of method to determine whether or not equally-sized axis-aligned disjoint cubes overlap each other when rendered.
Consider two opaque equally-sized axis-aligned disjoint cubes somewhere in three dimensional Euclidean space. Additionally there is a point in that space outside of the cubes that represents a camera ...
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An AF $C^*$ algebra has an approximate identity consisting of projections [duplicate]
Let $\mathfrak{A}$ be an AF $C^*$ algebra. Show that $\mathfrak{A}$ has an approximate identity $E_n$ (countable since this is an AF algebra) consisting of an increasing sequence of projections.
My ...
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Closure of the inverse image under the projection map
Let $S$ be a subsemigroup of a semitopological semigroup $(T,+)$, let $e$ be an idempotent in $T\setminus S$ such that $e\in cl_T(S)$, let $\mathcal{E}$ be a subsemigroup of $S\times S$ such that $(e,...
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The transformation matrix of an dimetric projection onto the plane Z=0
I need to make a dimetric projection of the point onto the Z=0 plane. I found the transformation matrix of an isometric projection onto the Z=0 plane and it looks like this:
\begin{align*}
M &= \...
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Sum of orthogonal projection operators in a Hilbert space
Let $\mathcal{H}$ be a Hilbert space and let $\{P_{\lambda} \ | \ \lambda \in \Lambda \}$ be a family of mutually orthogonal projection operators. I want to show that there exists a projection ...
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Orthogonal projection to the unit line (1,1,1,1) 4d
I have data that shows the strength of an effect for 4 conditions over multiple features. (For each Features the Effect are always the same sign)
I want to find a way to order the features by the ...
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If $x \in H$, show that $P_K(x)$ is characterized by the relations $(x, P_K(x)) = \Vert P_K(x)\Vert^2$ and $(x - P_K(x), y) = 0$ for all $y \in K$
Let $H$ be a real Hilbert space, let $K$ be a closed convex cone in $H$ with vertex at the origin. If $x \in H$, show that $P_K(x)$ is characterized by the relations $(x, P_K(x)) = \Vert P_K(x)\Vert^2$...
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Doubts about the form of the Euclidean projection
I've been studying some articles on the Generalized Alternate Projection (GAP) algorithm recently, but have a little doubt about the derivation of this algorithm.
In paper Generalized Alternating ...
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Product of projection operators
Let $H$ be a Hilbert space, let $E$ and $F$ be closed subspaces of $H$, and let $P: H \to E$ and $Q: H \to F$ be projection operators. Prove that $PQ$ is a projection operator if and only if $[E \...
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Given orthogonal projections P and Q on a Hilbert space such that norm of P-Q is less than one. Then rank of P and Q w.r.t. Hilbert dimension same.
Given two orthogonal projections P and Q on a Hilbert space such that $\|P-Q\|<1$. Then dim(range(P))=dim(range(Q)) w.r.t. Hilbert dimension.
Please note that the definition of Hilbert dimension ...
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Projection onto a subspace problem
I want to project a vector in $x-y$ plane to a one dimensional subspace $M$. $M$ is defined as follows:
\begin{equation}
M = \{(x,y):(x,y)=\alpha(-\cos\theta,\sin\theta), \alpha\in R \}
\end{equation}...
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Projection non-expansive in $\ell_1$ norm
It is well-known that for a convex closed set $K\subset\mathbb{R}^D$, the projection operator $\Pi:\mathbb{R}^D\rightarrow K$ given by
$$
\Pi(x)=\arg \min_{y\in K} \| x-y\|_2
$$
is non-expansive in ...
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Projecting an ellipse defined on a sphere onto the XY plane
I want to find the equation for and the area of the ellipse defined by $\Theta_H$ and $\Psi_H$ when it gets projected onto the XY plane.
The following diagram shows these ellipses.
Let us suppose we ...
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Let $U,W$ be finite-dimensional subspaces. Prove that $P_U P_W =0$ if and only if $\langle u,w\rangle = 0$ for all $u\in U, w\in W.$
This is question 6, page 201 of "Linear Algebra Done Right" by Axler (3rd Edition), which is a homework problem for my Linear Algebra course.
Suppose $U$ and $W$ are finite-dimensional ...
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Projecting orthogonal matrix onto a low rank subspace gives orthogonal matrix?
Let $U$ be an $n\times n$ uniformly-random orthogonal matrix, and let $P$ be a projection matrix onto a subspace of rank $k<n$.
Say I take the first $m\ll k$ columns of $U$, denoted as $U_1,\cdots,...
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If design matrices $X$, $Y$ are close to each other, are their projection matrices also close?
Let $A, B$ be two $m\times n$ matrices with $n > m$ and full row-rank. Define the following two (projection) matrices
$$
\begin{align}
P_A &= A^\top(AA^\top)^{-1}A \\
P_B &= B^\top(...
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Projection onto a hyperplane of the intersection of convex sets. [closed]
Let $A, B \subseteq \mathbb{R}^3$ be convex sets. More precisely, $A$ is a convex cone, pointed in $0$, while $B$ is a ball with centre in $0$ and finite radius $R$. Clearly, $A\cap B$ is non-empty. ...
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Projection of a Gaussian random vector onto the unit $\ell_1$ ball
Let $Z_n \in \mathbb{R}^n$, $Z_n \sim N(0, I_n)$ be a gaussian random vector, where $I_n$ is the identity matrix. The unit ball is defined as
$$ L_1 = \left[X \in \mathbb{R}^n: \| X \|_1 \leq 1 \...
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Showing that the operator $T=\sum_{n\in J}f(n)P_n$ is bounded and finding its spectrum in a Hilbert space $H=\bigoplus_{n\in J}H_n,P_n$ projection map
I have three questions related to the boundedness, operator norm and spectrum of the following operator in a Hilbert space $H$. I have bolded my exact questions for clarity.
Let $H$ be a Hilbert space ...
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When is proximal gradient descent better than projected gradient descent?
Suppose one is given two optimization problems ($P_1$ and $P_2$ respectively):
$$
\min_{z\in \mathbb R^n} \left\{\frac{1}{2} \|b - Az\|_2^2 + \iota_{\mathcal{C}}(z)\right\}
\\
\min_{z\in \mathbb R^n} \...
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