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.
1,458 questions
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SageMath: Orthogonal projection of $\mathbb{C}^3$ onto a subspace.
Consider the operator $T(x,y,z)=(x-iy+iz,ix-z,2y).$ I want to compute the orthogonal projection of $\mathbb{C}^3$ onto im$(T)$ using the projection formula $$P(v)=\langle v,w_1\rangle w_1+...+\langle ...
- 1,392
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Which non-unital $C^*$-algebras contain the *ortho-complements* of each of its projections?
Suppose $\mathcal{H}$ is a complex Hilbert space and $E$ is a projection in $\mathcal{B}(\mathcal{H})$. Then $E^\perp:=I_{\mathcal{H}} - E$ is the largest projection such that $EE^\perp = E^\perp E=0$....
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Verifying a spectral integral equality
I'm stucked when reading Chemin's Mathematical Geophysics An introduction to rotating fluids and the Navier-Stokes equations. On page 38 he applied spectral theorem to Stokes operator $\mathcal{A}$, ...
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Why are the projections of an area element onto the coordinate planes the components of its normal vector? [duplicate]
I understand that the components of the cross product $\bf{u}\times\bf{v}$ are the area of the parallelogram enclosed by $\bf{u}$ and $\bf{v}$, projected onto the respective coordinate planes (e.g. ...
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Do relational projections modify attributes?
Wikipedia gives a formal definition of a projection in relational algebra:
$$\Pi_{a_1, \ldots, a_n} = \{ t [a_1, \ldots, a_n]: t \in R \}$$
where $t [a_1, \ldots, a_n]$ is the restriction of the ...
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40
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Can the image of an everywhere rank zero map have dimension one?
Let $M \subset R^n$ be an $m$-dimensional properly embedded submanifold, and denote by $\pi_1$ the projection $M \rightarrow R$ of $M$ into its first coordinate. Assume I know that the rank of $\pi_1$ ...
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Why is $dA_{\perp} = dA\cos(\theta)$?
I'm sure there's an easy fix to this, but I can't seem to find it. Let $dA$ be a tiny patch of area on an arbitrary surface. Let $dA_{\perp}$ be the part of this area that is parallel to the vector ...
- 353
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Almost Projection open map
Consider the map $g:\mathbb{R}^2\times\mathbb{R}^2\to\mathbb{R}^2$, defined by
$$
g(x,y)=\begin{cases}
\overline{0},\qquad\quad \text{ if }x=\vec{0},\\
y^{\perp_x},\qquad \text{ if }x\neq\vec 0,
\end{...
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Map from product spaces is open
Let $X,Y$ be topological spaces and let $f:X\times Y\to X\times Y$ be a map such that $\pi_1\circ f=\pi_1$, where $\pi_1(x,y)=x$ is the projection on the first topological space. Suppose that, for any ...
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Low rank approximation to two matrices simultaneously
I'm trying to do a total-least squares-like regression but the set-up is slightly different and I end up with the following problem.
Let $C_1,\ C_2 \in \mathbb{R}^{n \times (d+1)}$ where $n >> d$...
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Different projection matrices, don't understand how to proceed
I'm stuck on the second part of an exercise:
In $\mathcal{E}^2$, identified with $\mathbb{R}^2$ through an orthonormal reference system $(i, j)$, be Pr: $\mathcal{E}^2 \to \mathcal{E}^2$, $v \mapsto v'...
- 629
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Represent a vector using a different orthonormal basis
I'm currently reading a book about quantum computing where it is important to represent a vector $|\psi⟩$ with $\| |\psi⟩ \| = 1$ with any given orthonormal basis. Given that $ \{ |b_1⟩, |b_2⟩, ... , |...
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$UU^*$ by columns
Background
Let $U\in \mathbb(C)^{n \times m}$ with $m<n$ have the columns $u_k$. Namely,
$$U = \begin{pmatrix}
\vert & & \vert \\
u_1 & ... & u_m \\
\vert & &...
- 381
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Proximal normals to a closed convex set.
I am interested in understanding the following infinite convex program:
$$ \underset{x \in R^{d}}{min} ||x-v||^2
$$
subject to the following infinite set of constraints:
$$ \langle \beta ,x \rangle \...
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Definition of sum of a family of mutually orthogonal projections
Suppose that $(P_i)$ is a family of mutually orthogonal projection acting on a Hilbert space $H$. The family need not to be countable in general. What is the definition of $\sum_i P_i$? More precisely,...
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Does $\arg\min_{x \in C}g(x) = P_C (\arg \min_{x}g(x))$?
Let $C$ be a convex set, $g(x)$ a convex function, and $P_C(x)$ is the projection map onto $C$. Is the following statement true?
$$\arg\min_{x \in C}g(x) = P_C (\arg \min_{x}g(x))$$ If the statement ...
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Planes attached point by point produce plane bundles
The issues I met came from quaternionic functions, but since the algebra is not involved I would rephrase it considering only the linear structure $\mathbb{R}^4$ or $\mathbb{R}^8$. I would like to ...
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How to obtain the orthogonal basis of an ellipsoid section
Given an ellipsoid equation of the form \begin{equation} \textbf{v}^T \textbf{A} \textbf{v} \; = \; 1 \tag{1} \end{equation} where $ \textbf{A} \in \mathbb{R}^{n \times n} $ is positive definite and ...
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Projection that varies point by point is an open map
Let us consider the vector space $\mathbb{R}^6=\mathbb{R}^3\times\mathbb{R}^3$ and let us represent any vector as $(x,y)$, with $x,y\in\mathbb{R}^3$. For fixed $x\in\mathbb{R}^3\setminus\{0\}$, let $y^...
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When does a projection preserve right-handedness?
Let $p = \begin{bmatrix}x \\ y \\ z\end{bmatrix}$ be a point in $\mathbb R^3$ which we wish to project to a point $q$ in $\mathbb R^2$. We can do this via a $2 \times 3$ matrix $A$ such that $$q = Ap....
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When is the product of two orthogonal projectors an orthogonal projector?
Let $V$ and $W$ be two vector subspaces of $\mathbb{R}^n$, let $p_V$ be the orthogonal projector onto $V$, and let $p_W$ be the orthogonal projector onto $W$. When is it the case that $p_Vp_W$ is the ...
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Is a bijective map from the set of complex numbers to the set of real numbers possible? [duplicate]
So, is there a bijection from $\mathbb{C}$ to $\mathbb{R}$?
A possible way: complex plane => sphere => square => segment => line.
The question is how to include all points along the path.
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Projection Operators and Tensor Product
I need some hints or ideas that I could use to solve this question:
Q) Let $A, B$ Idempotent, Hermitian, Linear Operators.
a) Firstly, show that: $\prod_A^a = \frac { (-1)^aA+1_V }{2} $ is a project ...
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Isometric conversion from screen coordinates to an $N \times M$ grid when not at true origin?
Given a pre-rendered image and its associated 2d array of $N \times M$ size, is there a system of equations that can calculate from screen $(x,y)$ to $(\text{grid}x, \text{grid}y)$ in my array? The ...
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Calculate error for an arbitrary projection matrix
I am attempting to find the best linear projection from a small set of $r,g,b$ coordinates to a $u, v$ coordinate space with the following characteristics:
$ 0 \le u \le 1 $
$ 0 \le v \le 1 $
The ...
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Sinkhorn Knopp algorithm - Bregman projection in update rule
I don't understand the updating rule for $u^{l+1}$ in the Sinkhorn algorithm. The below images contain all necessary definitions of the projection operators $A_1$ and $A_2$, which project a discrete ...
- 143
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Projection Methods in Sparse Matrices
Let $A$ be a symmetric positive definite matrix and fix $b \in \mathbb{R}^n$. Consider solving for the solution $x^* = A^{-1}b$ by using a projection method. In other words, we have a sequence of ...
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1-norm of projector matrix
Let $\mathbb R^{d\times k}$. It is well-known that the orthogonal projection operator over the column space of this matrix is given by
$$Px=A(A^\top A)^{-1}A^\top x.$$
Indeed, the 2-norm of this ...
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How to visualise why row addition doesn't change the determinant? [duplicate]
Geometric Interpretation of why row/column elementary operation doesn't change determinant?
I found this comment, which might potentially answer the question, but I'm not sure how to interpret it.
...
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Does the projection onto invariant subspace commute with one-parameter unitary group
Consider a Hilbert space $\mathcal{H}$ and a continuous one-parameter group of unitary transformations $\{U_t \}$ acting on it. Consider also the projection $\mathbb{P}$ onto the subspace of $\...
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Orthogonal Projection onto Half Space of Matrices
Given a matrix $\boldsymbol{Y} \in \mathbb{R}^{m \times n}$ how can one solve the orthogonal projection:
$$\begin{align}
\arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{X} - \...
- 9,470
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Embedding $\mathbb R[x,y]$ in the ring of functions $\mathbb N \times \mathbb N\to \mathbb R$.
I was attending a guest lecture at a university... We were discussing various problems. I was amused by the solution of this:
Let $R$ be the ring of all functions from $\mathbb N \times \mathbb N\to\...
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Lipschitz constant of projected function
Suppose $f:\mathbb{R}^n\longrightarrow\mathbb{R}^n$ is Lipschitz with constant $K$. Suppose I project $f$ onto orthogonal subspaces of $\mathbb{R}^n$, say $X$ and $Y$, where $\mathbb{R}^n= X \oplus Y$ ...
- 1,142
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Matrix Projection to Pseudo-Unitary Group
I'm trying to find the projection of a given matrix, $\mathbf{Y}\in\mathbb{C}^{2N\times 2N}$, to the pseudo unitary group. The problem can be formulated as follows:
\[
\min_{\mathbf{X}}\left\Vert\...
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Derivative of (squared) distance function [duplicate]
Consider some closed and convex set $A\subset\mathbb{R}^m$, where $m\in\mathbb{N}$. I am looking for a reference (in form of a paper or a book) concerning the differentiability of the distance ...
- 549
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uniqueness of a projection of a vector
Let $W$ a $k$-dimentional subspace of a finite dimensional inner product vector space $V$. If $v\in V$, then it is well known that the vector
$$
p=\langle v,w_1\rangle w_1+\ldots+\langle v,w_k\rangle ...
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Identifying the 3 Euler angles of rotation from 2 perspective images of a rectangle
I've worked on this problem for 3 days and I've decided to reach out for help. I have 2 images of the same rectangle (see figures). In the first image I have a 2D rectangle, and I know its height and ...
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The orthogonal projection to $W_1\cap W_2$ is given by $A_1A_2=A_2A_1$. What is the domain of this projection? ("Linear Algebra" by Ichiro Satake.)
I am reading "Linear Algebra" by Ichiro Satake.
The author wrote "The above proof also shows that the orthogonal projection to $W_1\cap W_2$ is given by $A_1A_2=A_2A_1$".
I don't ...
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Can the spinning dancer illusion be analyzed mathematically?
This is a type of Optical Illusion known as the Spinning Dancer Illusion:
It’s a famous optical illusion that many in this ...
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Understanding the Matrix $V^T(VV^T)^{-1}V$ and Its Minimum Element
Question:
Hi everyone,
I've been working with the matrix $M=V^T(VV^T)^{-1}V$, where $V$ is a $2\times n$ matrix with columns being 2D vectors $v_i$ whose barycenter $\frac{1}{n} \sum_jv_j$ is at the ...
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Equirectangular projection with multiple fisheye
I am trying to extend articles I found at https://paulbourke.net/dome/dualfish2sphere/ to project multiple frames from several fisheye cameras with partial overlap and 360 total FOV (with known ...
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Fisheye to equirectangular
I struggle to understand conversion from fisheye to equirectangular based on https://paulbourke.net/dome/dualfish2sphere/.
The first step of taken fisheye image coordinates $x$ and $y$ and ...
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Proving projection bound using Cauchy-Schwarz inequality
I'm reading a book The Cauchy-Schwarz Master Class Book by J. Michael Steele. And on page 59 (first paragraph), I found the assertion that we can prove the projection formula
$$
P(v) = tx = \frac{\...
- 143
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Determinant of a projection map composed with another linear map
I am reading MasCollel book "The theory of General Economic Equilibrium" and it contains a short review of key mathematical concepts used further in the theory. He states the following claim ...
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If $A\subseteq B$, then $B^{\perp} \subseteq \Pi(B\mid A^{\perp})$
Suppose A, B are closed linear subspace of Hilbert space, I want to show if $A\subseteq B$, then $B^{\perp} \subseteq \Pi(B\mid A^{\perp})$, where $\Pi$ is the projection operator.
I'm not sure how to ...
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$L^2$ vs $L^\infty$ projection
Let $\mathbb P_N$ the space polynomials of degree at most $N$ on $X=[-1,1]$. What is
$$\sup_{f\in L^\infty(X)\setminus \mathbb P_N}\frac{\|f-P_2[f]\|_{\infty}}{d_\infty(f,\mathbb P_N)},$$
where $d_\...
- 1,207
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How to project one circle onto another for the purposes of angles?
Two circles are concentric with the same radius. Circle 2 (blue) is tilted relative to the xy plane. They coincide at some nodal axis (dotted line) which is the x axis.
Now we advance to some point A ...
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$(Px,x)=\|Px\|^2$ implies that $P$ is a projection.
Let $H$ be a Hilbert space over $\mathbb{R}$ and $P$ is a bounded linear operator over it. If
$$
(Px,x)=\|Px\|^2,\quad \forall x\in H,
$$
I want to show $P$ is a projection.
If $H$ is a Hilbert space ...
- 367
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Orthogonal Projection onto a Polyhedron (Matrix Inequality)
How to efficiently solve:
$$\begin{align*}
\arg \min_{\boldsymbol{X}} \quad & \frac{1}{2} {\left\| \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} \\
\text{subject to} \quad & \begin{aligned}...
- 9,470
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Characteristic polynomial of an orthogonal projection
Q: What is the characteristic polynomial of an orthogonal projection onto a (two-dimensional) plane through the origin in $\mathbb R^4$?
Ans: $x^2(x-1)^2$
Can someone please explain how to do this ...
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