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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Proving “annihilator matrix ”is positive semidefinite

My question is about "proving that $\mathbf{I} - \mathbf{X} ( \mathbf{X}' \mathbf{X} )^{-1} \mathbf{ X }' $ is positive semidefinite?" For instance, I attempted proving that $$\mathbf{c'} [\mathbf{I} ...
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How would I go about solving this projection matrices question?

The linear algebra course that I'm taking has provided us with examples of how to find the orthogonal projection of a vector onto a subspace given the orthogonal basis of that subspace, however, I am ...
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45 views

Orthogonal projection matrix of a matrix with one column sign switched

Suppose that I have a $n \times k$ real matrix with full column rank. Say $k=3$ and I write $$X = [\mathbf x_1:\mathbf x_2:\mathbf x_3]$$ where lower-case $\mathbf x$'s are $n \times 1$ vectors. I ...
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Conditions for conmutative projection matrices

Assume you have a vector $x$ $\in {\rm I\!R}^n$ and two projection matrices (that only differ with respect to the metric that defines the inner product) $P_1=S^{}(S'W_1^{}S^{})^{-1}S'W_1^{}$ and $P_2=...
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Ortogonal basis for a subspace of $R^{4}$ and find a point in a plane closest to the origin.

Let $V \subset R^{4}$ a subspace defined for the equation $x_{1}+3x_{2}-5x_{3}-x_{4}=0$. a) Find a ortogonal basis for $V$. b) Wich point in the plane $x_{1}+3x_{2}-5x_{3}-x_{4}=36$ is closest to $(...
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In the case below, calculate (i) the orthogonal projection $ P_W $ associated with the two-dimensional vector subspace W = L (S ') of the …

In the case below, calculate (i) the orthogonal projection $ P_W $ associated with the two-dimensional vector subspace W = L (S ') of the vector space V generated by linearly independent set S '= $ \{\...
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31 views

How do I write this equation backwards (with inverted matrices)?

My question is what matrix operations are used to get to from eq.(1) to eq.(3)? I'm pretty clear with eq.(1), solving for the unknown image pts [u,v]. Now I want to solve for an unknown XYZ world ...
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30 views

Why is the error vector $e = b - p$ ( Introduction to Linear Algebra of Gilbert Strang)

I have a question in the projection parts (chapter 4 in his book). I don't really understand why $e=b-p$ (when $e$ is error vector). A line goes through the origin in the direction $a=(a_1,...,a_m)$. ...
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What is the projection matrix of reverse (Byzantine) perspective?

I would like to construct a projection matrix for reverse perspective. I'm using OpenGL and tried to modify concepts from this excelent tutorial. I came up with: $$ \begin{bmatrix} 2\frac{(\text{near}...
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How does one calculate the projections onto sub-spaces of $C^3$?

Given the following two sub-spaces of $C^3$: $W = \operatorname{span}[(1,0,0)] \,, U = \operatorname{span}[(1,1,0),(0,1,1)].$ I want to find the linear operators $P_u , P_w$ which represent the ...
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Finding Matrix of Projection

Let $P:R4→R4$ be the orthogonal projection onto the plane $ W={−(y+2z+t)=0}$ (That is, the projection parallel to the normal vector $(0,−1,−2,−1)$.) Find the matrix $M^{E}_{E} (P)$ of ...
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What is this projection matrix doing?

Let’s say we have a $m\times d$ zero mean multivariate Gaussian matrix $X$. Its covariance matrix is $X^{T}X$. Let $V$ be the $d\times d$ matrix of eigenvectors of $X^{T}X$, with the columns sorted in ...
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Infinite number of projectors

I am studying optics and I got to te part of the matrix treatment of polarization. Here I'll be talking only about linear polarization, hence it just a mere treatment of projectors in two dimensions. ...
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Orthogonal Projections Are Symmetric - Geometric Intuition

Let us denote the projection matrix onto the column space of $A$ by $\pi_A = A(A^T A)^{-1} A^T$. I am looking for geometric intuition as to why it is symmetric. It is very clear to me due to plenty of ...
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Compute the projection of a function in a different way

The problem: Let C([0,1]) denote the vector space of all continuous real-valued functions defined on the closed unit interval [0,1]. For $f, g \, \, \in\, C([0,1])$ we define the inner product: $$ (...
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57 views

Trace, dimension and equivalence in $M_n(\mathbb{C})$

Let $tr: M_n(\mathbb{C}) \to \mathbb{C}$ be the standard trace given by $tr \begin{pmatrix} x_{11} & x_{12} & x_{13} & \dots & x_{1n} \\ x_{21} & x_{22} & x_{23} &...
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How do I solve for the projection of a vector on one axis mathematically?

If I have a vector $v$ = (x1, x2) and I want to find the matrix A so that I may express the projected vector on an axis (Y=AX), how would I do this mathematically? If there is a line through the ...
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Sensitivity of homographic image to movement of 4 defining points

A transformation $T:\mathbf{p}\mapsto \mathbf{p}+\begin{bmatrix} x \\ y \end{bmatrix}$ is defined by 4 given anchor points in $\mathbb{R}^2$, and their images in $\mathbb{R}^2$: $$ \mathbf{p}_0 \...
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48 views

How to construct a projection matrix to the midpoint of two points with rotation

Picture of Scenario I have an initial and terminal point. I have a unit cube located at the origin. I want to construct a projection matrix to move the cube to the origin, rotate, and scale it so ...
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Can we compare class separability ratio from two different Fisher LDA subspaces?

I am working on a c-class problem separation with d-dimensional samples. I use the Fisher Discriminent Analysis for dimension reduction. This approach gives a subspace based on w that maximizes the ...
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34 views

Projection of a vector being spanned

Find the Projection of [2,4,1,3] on to W spanned by {[.7,.1,.7,.1],[.7,.1,-.7,-.1]}. What would be the equation to start the process? Is it (((y*v1)/v1*v1))v1)+ (((yv2)/v2*v2))*v2)
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Inclusion of subspaces

Consider full column rank (injective) matrices $A \in \mathbb{R}^{n\times q}$ and $B\in \mathbb{R}^{n\times d}$ for some $n\geq d,q\geq 1$. Furthermore, assume that $A^T B$ is of full column rank ...
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Making a homogeneous projection matrix that changes perspective while maintaining the plane at z=0 invariant?

I'd like to achieve an effect that looks like this (mockup in blender) As you can see, everything at the z=0 plane stays in place. I tried doing this: let P be the standard perspective projection ...
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32 views

Two matrices such that AB=B and BA=A yield collinear projections?

$A$ and $B$ are two matrices such that $AB=B$ and $BA=A$. For vector $z$, let $x=Az$ and $y=Bz$. Can we show two vectors $x$ and $y$ are in the same direction? Is there any special condition required ...
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How can I prove that if $M$ is a projection matrix onto the column space of $X$, $C(X)$, then $C(M) = C(X)$?

I came across the claim that if M is a perpendicular projection matrix onto the column space of $X$,$C(X)$, then $C(M) = C(X)$. I wanted to prove this claim but I am not sure my argument is correct. ...
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9 views

Sum of orthogonal projections onto orthogonal subspaces, where orthogonal subspaces forms a direct sum

I wish to prove, if possible, that if I have a set of non-zero orthogonal subspaces of $\mathbb{R}^n$, $U_1,\ldots,U_m$, with orthogonal projections $U_i,\ldots,U_m$ respectively, where $\mathbb{R}^n =...
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59 views

Can a homography transform a straight-line into another type of curve?

I am trying to find the homography that maps a certain image to another related image. This can be done by selecting at least $4$ point-to-point correspondences, then we can solve a constrained ...
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42 views

Projection onto orthogonal complement as product of projections

Consider a matrix $A = \begin{bmatrix} A_1 & A_2 \end{bmatrix}\in \mathbb{R}^{n\times(d_1+d_2)}$ with $rank(A) = d_1+d_2=d\leq n$. That is, the columns of $A$ are linearly independent. I want to ...
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Othogonality and bias in regression

I am reading a text where given an expression for omitted variable bias $$(1) \ \hat \lambda = \lambda + (D^\top M_XD)^{-1} D^\top M_X F\psi$$ it is stated that if $D$ is orthogonal to $X$ such that $...
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A neat proof of equality involving projection matrices that is reminiscent of Cauchy-Schwarz

Trying to prove: $$ \|X^\top(I - H_0)Y\|^2= \|(H-H_0)Y\| \|(I - H_0)X\| \tag{1} $$ Here's where this expression comes from. In a linear regression $$ Y = X\beta + \varepsilon, $$ I define two (...
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Cut of the cube projected to the rectangle

I have a 3D euclidean space. Somewhere in this space is positioned unit cube, that is aligned with the axes of the 3D geometry. Assume that there is arbitrarily placed 2D square in the space that must ...
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1answer
67 views

Convex hull and projective matrices

If $\mathcal{A}$ is a unital $C^*$-algebra then I want to show that $\operatorname{conv}(\operatorname{Proj}(\mathcal{A}))=\mathcal{P}_1(\mathcal{A}):= \lbrace x \in \mathcal{A^+} : \Vert x \Vert \...
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Understanding a step in the characterization of projections

I was reading the paper: Unitary similarity of projectors by Dragomir Z. Dokovic. I do not follow a certain step. The setting is this: $p$ is a projection (not necessarily orthogonal) in a finite ...
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How to do point decomposition with projections

I am taking a linear algebra class for data science reduction techniques. It has been a few years since I took linear algebra and I am struggling on some of the terms and processes. I need to ...
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38 views

Perspective text convert to Math Language

Rule number one of perspective is that objects of the same real size, look smaller the further away they are from the viewer. Therefore a man of 6ft who is standing about 6 feet away from you, will ...
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Attempting to project 4 values, to 5 values points to calculate an approximate average of 4 point values in 5 point value space.

Attempting to project 4 values, to 5 values points to calculate an approximate average of 4 point values in 5 point value space. The problem: Attempting to map a 4 point proficiency grading to a 5 ...
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1answer
39 views

Orthogonal Projection of Approximate Matrix

Given a matrix $A$ its orthogonal projection is given by $P_A = (AA^T)^{-1}A^T$. Now, assume we know $A \in B_{\epsilon}(\bar{A})$, that is $||A - \bar{A}||_2 \leq \epsilon$. Here $\bar{A}$ is some ...
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relationship between focal length, perspective projection and camera distance

This page here describes that the focal length does not result in perspective distortion: "Yes, there are optical distortions in lenses, but those have nothing to do with the focal length." while ...
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Compactness of the set of rank K projectors

I hope you could give a hint for proving that the following set is compact $(k<n)$: $X=\left\{A\in \mathbb{R}^{n\times n}:A=A^{t},A^{2}=A,rank(A)=k\right\}$ I can proof that $X$ is bounded(not so ...
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Find the orthogonal projection $p$ of the vector $v$ on the line $\{ta+b, -\infty < t < \infty\}$

This is a practice problem in a 400-level linear algebra course. This is supposed to be review of 200-level material. For some reason, I'm having a mental block trying to understand the solution given....
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How do you write the projection operator of a direct sum in terms of basis elements?

Let $U$ and $V$ be two subspaces of $\mathbb{R}^n$ satisfying $U\cap V=\{0\}$, and let $P$ be the projection operator from $U+V$ to $U$. My question is, is there a way to explicitly compute $Px$ for ...
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Prove that Px1*Px = Px1, and Mx1*Mx = Mx

Consider the following linear regression model: y = Xβ + u = X1β1 + X2β2 + u where y and u are n × 1 vectors, X1 and X2 are n × k1 and n × k2 matrices of explanatory variables. Define the following ...
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Does a projection matrix have to be a square matrix?

Does a projection matrix have to be a square matrix? I know that it's computed by a formula $$P = A(A^TA)^{-1}A^T$$ where $A$ can be virtually of any dimensions. Does this formula guarantee that $P$ ...
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Subspaces of the projection - matrix

I have some problems with the following exercise: Let $v_1 = [1,1,0]^T,v_2 = [1,0,1]^T, v_3 = [0,0,1]^T$. Find the dual basis $\bar{v}^1,\bar{v}^2,\bar{v}^3$ such that $\bar{v}^{i}(v_{j})= \delta^{i}...
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Since $~P(X) ~\bot~ x-P(X)~ $, we have $~||x||^2=||p(x)||^2+||x-P(x) ||^2~$

"Since $~P(X) \ \bot ~\ x -P(X) ~$, we have $~||x||^2=||p(x)||^2+||x-P(x) ||^2~$ ". Can someone please explain why this is true? Since $~P(X) \ \bot ~\ x -P(X) ~$, we have $~||x||^2=||p(x)||^2+||x-...
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105 views

Transpose of a matrix and the product $A A^\top$

I've been following 3Blue1Brown's Essence of Linear Algebra, basically the question (1) is what is the geometric meaning of the transpose? I've watched Chapter 9-Dot Products and duality . I can see ...
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On the null spaces of projection operators having the same range

Let $K$ be a field of characteristic zero. If $T,S: K^n \to K^n$ be linear operators such that $T^2=T, S^2=S$ and range$(T)=$range$(S)$, then is it true that $\ker T=\ker S$ ? Since $T,S$ have ...
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Orthogonal Subspace: $A_p$ and $B_p$ are orthogonally projected by the same projector $P$. Is there always $P$ so that $A_p=QB_p$, where $Q\in O(k)$?

Orthogonal Subspace: $A_p$ and $B_p$ are orthogonally projected by the same projector $P$. Existence of $P$ always so that $A_p=QB_p$, where $Q\in O(k)$? Let $A,B\in \mathbb{R}^{k\times n}$ with $A\...
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49 views

Optimal low-rank approximation of a matrix for given basis

Given a target matrix $\mathbf{T}\in \mathbb{C}^{M\times M}$ and a basis (not necessarily orthonormal) $\mathbf{V}\in \mathbb{C}^{M\times K}$, how would I find the optimal reconstruction of $\mathbf{T}...
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58 views

Null Space and Image of Projection

I'm having trouble solving this question: Consider in a vector space $V = M(3 × 3, \mathbb{C})$ of complex $n \times n$ matrices with inner product $\langle A, B\rangle = tr(A^∗B)$ the projection $P :...

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