# Questions tagged [projection-matrices]

This tag is for questions relating to projection matrix, which is an square matrix that gives a vector space projection from to a subspace.

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### Can every lattice obtained by orthogonal projection of the cubic lattice?

Assume that $\Lambda\subset\mathbb{Z}^n$ is a full-dimensional lattice with the generator matrix $G\in\mathbb{Z}^{n\times n}$. Can I always embed $\mathbb{R}^n$ into a bigger dimensional vector space, ...
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### If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2).$

Prove or provide a counterexample: If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2),$ where $\mathrm{tr}$ denoted the trace and $\mathrm{rk}$ the ...
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### Project 'b' onto 'a' to find out the projection of 'b' onto 'a' [closed]

[Question: Project 'b' onto 'a' to find out the projection of 'b' onto 'a'. To solve this, which formula should be used to find out the projection? Scalar Projection or Vector Projection? Lil bit ...
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### Cross-ratio: Role in non-Euclidean geometry. Invariant for the action of $\text{Stab}(C)$ on pairs of points?

The Wikipedia explanation of the role of cross-ratio in non-euclidean Geometry. "There is an invariant for the action of $G_C$ on pairs of points," the article claims. What does it mean, ...
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1 vote
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### A question regarding to the Wikipedia definition of homography (or projective transformation).

According to the section Definition and expression in homogeneous coordinates of this Wikipedia article, we get \begin{align} y_1 &= \frac{a_{1,0} + a_{1,1}x_1 +\dots + a_{1,n}x_n}{a_{0,0} + a_{0,...
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### Oblique projection onto an intersection along to a sum of vector spaces.

(Sorry for the long post, this problem is a head scratcher.) Definition. Assume $\mathbf{R}^d = V \oplus W,$ in other words, we assume $\mathbf{R}^d$ is the direct sum of two of its subspaces, and by ...
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1 vote
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### Orthogonal projection of an ellipsoïd from N to 2 dimensional space

Suppose we have a $N\times N$ symmetric-positive-definite matrix $A$, representing an ellipsoïd in $N$ dimensional space. How to find the matrix $A_{xy}$ corresponding to orthogonal projection of ...
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### Orthogonal project of a vector $\vec{y}$ that sit in the same plane (aka span($u_1, u_2$)) such that $\vec{y},\vec{u_1}, \vec{u_2} \in \mathbb{R^3}$

Confirmed that $u_1,u_2$ are indeed orthogonal by taking their dot product which equals zero. Correct answer = $\begin{pmatrix}-1\\ -1\\ 6\end{pmatrix}$. Geographically, mapped out in geogebra: we see ...
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### Deriving projection matrix $QQ^Tb=\hat b$ from normal equation

Suppose normal equation $A^TA\hat x=A^Tb$, with $A$ has linearly dependent columns. Is it possible to directly derive from this normal equation that $QQ^Tb=\hat b$, with $A=QR$? Or we can only prove ...
1 vote
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### When does $PAP = (Q \otimes R) A (Q \otimes R)$ imply $P$ is a tensor product, for orthogonal projections $P$, $Q$, and $R$?

Suppose $A$ is a positive semi-definite operator on $V \otimes W$ such that $\operatorname{tr}(A) > 0$, where $V$ and $W$ are finite-dimensional complex vector spaces. Let $P$ be an orthogonal ...
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### Prove: affine transformation maps "line at infinity" to "line at infinity"

I'm studying Computer Vision and my lecturer stated that: The affine transformation maps "line at infinity" to "line at infinity". I'm trying to prove it as part of my ...
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### Projector that annihilates exactly those vectors that are annihilated by every projector in a set

Consider a finite-dimensional Hilbert space, and consider a set of not-necessarily-commuting projectors $\{P_i\}$ that act on the space. Consider the linear subspace of vectors $S$ that are ...
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### SVD of an orthogonal projector

Here is my observation: Suppose there is an orthogonal projector $P$ such that $P=P^2$. Then for arbitrary $x$, $Px$ and $(I-P)x$ are orthogonal. So we have $$x^* P^* (I-P)x=0$$ where $A^*$ means ...
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### Understanding orthogonal projection of $\vec{y}$ on to span of orthogonal set, with an example

This is to verify if there's an issue with my understanding or if there's issue with the textbook. There seem to be also a previous question here on exactly the same, hoping to help myself and future ...
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### a projective geometry question

if I understand correctly then, that the projective geometry is the geometry at which the only straight lines are preserved meaning that points stay points and lines stay lines and conics stay conics (...
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### Unitary operator times projection operator

In this paper, the authors claim that for $C$ a unitary operator and $P$ a projection operator, if $CP \propto P$, then the constant of proportionality must be one. I don't see why this must be the ...
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### Why does acting as the inverse on a subspace guarantee commutativity?

Let $V$ be some vector space and $P$ a projection onto a subvectorspace $C \subset V$. Let $\mathcal L$ be a set of unitary operators on $V$ satisfying $AP = PAP$ for all $A \in \mathcal L$. In this ...
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### Singular Values of Projection matrix

In this question, someone proves that singular values of projection matrix must larger than 1: A projection $P$ is orthogonal if and only if its spectral norm is 1 Is it a right rule of singular ...
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### Show that the relative error for points of a power regression $f=(f_{1}, f_{2},...,f_{m})$ can be found in $(e^{f_{1}}-1,e^{f_{2}}-1,...,e^{f_{m}}-1)$

You are given a series of x-points corresponding to a set of y-points. The first part to this task is to calculate the projection of the vector and determine the absolute error for these points by ...
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### Least squares projection with singular matrix

I am trying to better understand the solution of systems of linear equations (also in an $L_2$ sense for inconsistent ones) and for that purpose I have been trying to derive the set of solutions in ...
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### $\ell_1$ norm of random projection

Let $G_{m, n}$ denote the Grassmannian manifold, i.e. the set containing all possible subspaces of $R^m$ with dimension $n$. Let $E \in G_{m, n}$. We can associate with $E$ an orthogonal projection ...
1 vote
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