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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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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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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Eigenvalues of $AA^T$ if $A^T A=I_r$

Suppose $A$ is a $n \times r$ matrix with $n>r$ and $A^TA=I_r$, i.e., $A$ could be the matrix of eigenvectors of $r$ eigenvalues. I am wondering what's the eigenvalues of $AA^T$, which is of rank $...
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SDP problem with Schur complement-like constraint

I'm interested in what we can conclude about the following question. Let $\mathbf{Y} \in \mathbb{R}^{n \times n}$ be in the convex hull of projection matrices of rank at most $k$ (equivalently, $\...
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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 ...
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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 ...
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1 answer
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Prove rank $I - uu^* = n-1$ (using ... row reduction?)

Show that rank $(I - uu^*) = n-1$ where $\|u\| = 1$. Does the following proof go through? Row reduce $uu^*$ as follows: $$ \begin{bmatrix} u_1\bar{u}_1 & \cdots & u_1 \bar{u}_n \\ \vdots &...
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SVD of residual matrix non-orthogonal to orthogonal projection?

Suppose we have two data matrices, $X$ which is a $m$ x $n$ matrix, where $m$ >> $n$ $Y$ which is a $n$ x $p$ matrix, consisting of $p$ orthonormal columns, where $n$ > $p$ Next, we find ...
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Intersection of the null space of a matrix with integer vectors

I have an $m\times n$ rational matrix $A$ which is full rank and has linearly independent columns. With $m>n$ I would like to find all integer values of $x$ for which $$A\vec x=0$$ If $A$ was an ...
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Reducing the norm of a vector

Let $A$ and $B$ be a $n\times k_1$ and $n\times k_2$ matrices, and let $b$ be a vector in $\mathbb{R}^n$ (we may assume that $[1\ldots 1]^{\top}b=0$). We denote the projection matrix that sends a ...
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Spectral norm of matrix restricted to subspace $1^\perp$

On Page 6 (of 23) of the following document: Boyd et. al - Fastest Mixing Markov Chain on a Graph The spectral norm of a matrix P, restricted to subspace $\mathbf{1}^\perp=\{u \in R^{n} \mid \mathbf{1}...
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Deviation of Random Projections

Let $P$ be an orthogonal projection in $\mathbb{R}^n$ onto an $m-$dimensional random subspace uniformly distributed in the Grassmannian $G_{n,m}$. Let $T$ be a bounded subset of $\mathbb{R}^n$. Let $x$...
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1 answer
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Uniqueness of affine forms

Lets define the power symmetric group $$\text{psg}_{n,d}(x_1,\dots, x_n) = \sum_{i=1}^n x_i^d$$ And lets define a linear form as $$\ell_i(x_1,\dots,x_n)=\sum_{j=1}^{n} a_{ij}x_i+a_{i0}$$ Where $a_{ij} ...
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Homography matrix in camera calibration

I have been following the slides from uzh on camera calibration(https://rpg.ifi.uzh.ch/docs/teaching/2020/03_camera_calibration.pdf) with direct linear transform. On slide 33 they say that system ...
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What's the correct representation of projection matrix between $UU^T$ and $U(U^TU)^{-1}U^T$

I have seen those two representation of projection matrix $UU^T$ and $U(U^TU)^{-1}U^T$, where column of $U$ are orthonormal basis of subspace $S$. Which one is more accurate? For projection matrix $UU^...
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1 vote
1 answer
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Given a point in an image find another point that is a certain distance away when projected to world coordinates

I have a point in image space p1 = (u1, v1) where u and v are the pixel coordinates. I project this point out in to 3D coordinates, using the projection matrix of the camera, to intersect with a plane ...
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2 votes
1 answer
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Must a "projector" on a normed space be bounded?

In V. Moretti's Spectral Theory and Quantum Mechanics, a projector on a normed space $X$ is defined as a bounded linear map $P:X\to X$ such that $P^2=P$. Is the boundedness condition really required ...
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errors associated with each observations based on their distance to a linear regression plane

This is in reference to outlier analysis by Charu C Aggarwal. Let $D$ be a dataset of dimension $N \times d$ where N is the number of observations and d is the dimensions (or variables). Here, $D$ is ...
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1 vote
2 answers
158 views

Why is the pseudoinverse of an orthogonal projection matrix itself?

From both this paper and Wikipedia, it is mentioned that for an orthogonal projection matrix $(I - A^+A)$ its pseudo inverse is itself, i.e., $$(I - A^+A)^+ = I - A^+A$$ Why is this the case? Can ...
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How to find the P matrix when A is singular? Why does it work?

Inspired by this exercise in Strang's book: Suppose the columns of $A$ are not independent. How could you find a matrix $B$ so that $P = B( B^T B )^{-1}B^T$ does give the projection onto the column ...
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How to find the restriction of a semidefinite matrix to positive eigenvalues?

I have a Hermitian 4x4 matrix $H$ which has 2 positive and 2 negative eigenvalues with orthogonal eigenvectors. Let $P$ be the projector to the 2-plane given by the two eigenvectors with positive ...
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1 answer
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What makes inequality true in proof of Gauss Markov theorem

Elsewhere on this site, I found a very compact proof of the Gauss-Markov theorem, seen below. I don't understand the justification for the middle step with the inequality. Specifically, what property ...
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What function yields a projection matrix with the smallest Frobenius norm to a given projection matrix?

Phrasing Attempt 1 If I have one function $f_1: \mathcal{X} \rightarrow \mathbb{R}^{D_1}$ that yields a particular projection matrix $P_1 \in \mathbb{R}^{N \times N}$, how do I find the function $f_2: ...
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Following a calculation of entropy in the first quantization scheme

I'm trying to follow the computations of example 5.1 in this paper. To begin with they have a symplectic Hilber space $(\mathcal{K},\tau,\sigma)$, where $(\mathcal{K},\tau)$ is a separable Hilbert ...
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-3 votes
2 answers
242 views

Prove that projection matrix has the same rank as design matrix. [closed]

The equation for projection matrix is as follows: $$\begin{align} \mathrm{P = A\left(A^TA\right)^{-1}A^T} \end{align}$$ Here we want to project onto column space of $\mathrm A$. How to prove that $\...
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Is $pq+qp$ ever a projection?

There is a question over on Mathoverflow to which I added this answer. Is it an answer? i.e. if $p$ and $q$ are Hilbert space projections, $p=p^2=p^*$ and $q=q^2=q^*$, is $pq+qp$ ever a non-zero ...
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3 votes
1 answer
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What is the orthogonal projection with expectation?

I am reading an advanced econometrics textbook. When it talks about least squares, it says that the orthogonal projection of A onto Z is $P_Z(A)=Z^\prime E[ZZ^\prime]^{-1}E[ZA_k]$ and when A is a ...
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1 vote
1 answer
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Products of Non-Commuting (or orthogonal) Projections

Let $S=\{P_1,P_2,\dots,P_n\}$ be a set of Hilbert space projections $P_i=P_i^2=P_i^*$ that only commute when they are orthogonal or equal: $$P_iP_j=P_jP_i\implies P_iP_j=0\text{ or }i=j.$$ Let $i,j,k$ ...
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If $H$ is a projection matrix and $J$ is a matrix of ones, is it always true that $HJ = J$?

I'm trying to prove that if $H$ is a projection matrix and $J$ is a matrix of ones, is it always true that $HJ = J$? For example, $$ \begin{bmatrix} 0 & 1 \\ 0 & 1 \end{bmatrix} \begin{bmatrix}...
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Can I only use matrix multiplication to solve the Projection Area of 3D Shapes problem?

Let's say there are some 1 x 1 x 1 cubes that are axis-aligned with the x, y, and ...
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1 vote
1 answer
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The power equation in $\mathrm{PSL}_2(\mathbb{C})$

Problem. Show that the equation $$x^2=\begin{pmatrix}-1&1\\0&-1\end{pmatrix}$$ has no solutions in $\mathrm{SL}_2(\mathbb{C})$. From this, show that any power equation $x^m=a$ is solvable in $\...
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2 votes
2 answers
108 views

Prove that a projection matrix for a subspace is the basis matrix of the subspace times its transpose.

Let V be a real linear subspace and U be a matrix whose columns form an orthonormal basis for V. How can I prove that $$proj_v(x) = UU^tx$$ I've been struggling for linear algebra, so specific, ...
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1 answer
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Finding rotation applied on a 3D point so that it lands to a specific pixel

Suppose we have a 3D point $X=(x,y,z)$. Given the projection matrix of a camera: $$ \begin{matrix} f_x & 0 & c_x\\ 0 & f_y & c_y\\ 0 & 0 & 1 \\ \end{matrix} ...
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