# 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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### 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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### 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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### 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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### 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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### 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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