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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17 views

Parametric form of a cone

Given the following parametric form: $P(v)=P_1+(P_2−P_1)v$ where $P_1=(1,0,0)^T$ and $P_2=(0,0,1)^T$ with $0≤v≤1$ , what type of parametric object is $Q(u ,v)=R_z(u)P(v)$ if $0≤u<2π $ and matrix $...
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21 views

Show that each eigenvalue of matrix $M_Z \Sigma M_Z$ is less than the corresponding one of $M_X \Sigma M_X$

Let $Z:=[X \hspace{0.2cm} Y]$ be an $n\times k$ partitioned real matrix with rank $k$ and $$M_X:=I_n-X(X'X)^{-1}X'$$ $$M_Z:=I_n-Z(Z'Z)^{-1}Z'$$ If $\Sigma$ is an $n\times n$ positive semi-definite ...
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32 views

Reconstructing $3D$ point cloud from depth maps and known camera parameters

The Task I have four depth maps of the same scene, one from each side (front, back, left, right). Given a depth map, I would like to reconstruct a 3D point cloud. For each depth map I have the ...
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45 views

If $PP^*=P^*P$ and $P^2=P$ then show that $P^*=P$

To show an idempotent normal matrix is an $\textbf{orthogonal}$ projection. Proof: If we use, spectral decomposition of normal matrices we can write, $P=U\Lambda U^* \implies P^2=U\Lambda^2U^*$. ...
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20 views

Intuition behind estimable contrasts in linear regression with singular model matrix

I am struggling with developing an intuitive understanding of estimability in linear regression with a singular design matrix. Here is the setting I am talking about: Let $Y = (Y_1, ..., Y_n)' = Xb + \...
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23 views

Push-through rule for square root of specific projection matrix

Is the following a valid identity? $$ (I - XX^T)^{1/2} X = X (I - X^T X)^{1/2} $$ Without the square root, it is quite easy to see that $$ (I - XX^T) X = X (I - X^T X) $$ holds but I have trouble ...
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63 views

Rank of projection error

Let $X_1$,$X_2$ be $n\times k$ real matrices with rank $k$. Let $S_1,S_2$ denote the column spaces of $X_1,X_2$ respectively. I am trying to show that the rank of $MX_1 $, with $M:=I_n-X_2(X_2'X_2)^{-...
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31 views

Values for x for which distance between vector and subspace is maximal

Given $A=\left ( 1, 1, 1 \right )$ be vector in $\mathbb{R}^{3}$. Let $L$ be a subspace of $\mathbb{R}^{3}$ spanned by the vectors $\left ( 1, 0, -1 \right )$ and $\left ( 3, 5, x \right )$. Find the ...
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59 views

Are all Givens rotations linear transformations?

I've read that Givens rotations are linear transformations, but as we know linear transformations preserve the length of the vector transformed. However when I tried this with a 3 dimensional Givens ...
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20 views

Understanding orthogonal vector projections

I am trying to understand the linear algebra of PCA, and specifically what does it mean to be the $i$th principal component of some random vector $x$. I do know that this refers to the coordinate of $...
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24 views

Find projection to linear orthogonal subspace using linear regression when matrix is not invertible

Task: $X_1=(-2,6,3,-1)^T$ $X_2=(7,-3,-6,2)^T$ $X_3=(3,9,0,0)^T$ $Y=(2,-1,3,8)^T$ I need to find projection to Z, which is a linear projection. $Z$ should be expressed as $X_1,X_2$ and $X_3 $ ...
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19 views

Approximating $AB$ by $AP$ where $P$ is an orthogonal projection

Let $A\in {R} ^{m \times d}$, $B \in R^{d \times d}$. Given $n\leq d$, I want to find an orthogonal projection $P$ onto an $n$-dimensional subspace of $R^d$ s.t $AP$ best approximates $AB$ according ...
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19 views

Bound on eigenvalues for subset of columns of projection matrix

I'm taking the product of an orthogonal projection matrix (symmetric), $P$, with a diagonal matrix $D$ whose entries are either 0 or 1. So, $\tilde{P} = PD$ results in zeroing-out some of the columns ...
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36 views

Definition and Doubt: Universal $\mathrm{C}^*$-Algebras given by generators and relations

Let $A:=\mathrm{C}^*(G,R)$ be the (defined) universal $\mathrm{C}^*$-algebra generated by generators $G$ and relations $R$. As far as I can discern, there are two ways of defining such objects --- ...
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49 views

Computing the distance between vector and its projection on a random subspace

Let $V\in\mathbb{R}^{n\times m}$, where $n>m$, be a random matrix of standard normal gaussians. Given an arbitrary vector $y\in\mathbb{R}^n$, I need to understand the distance $||y-p_V(y)||^2$, ...
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28 views

How to find the orthogonal projection of a matrix onto a subspace?

$ \newcommand{\<}{\left \langle} \newcommand{\>}{\right \rangle} $ I have fought 2 vectors(?!) in basis which are $M_1 = \begin{bmatrix} 1 & 0\\ 0 & -1\\ \end{bmatrix}$ and $...
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1answer
47 views

Rank of eigenvector projections

Consider the following two $n\times n$ matrices: $$A_1:=H_1\Sigma_1H_1' $$ $$A_2:=H_2\Sigma_2H_2'$$ where $\Sigma_1$,$\Sigma_2$ are positive definite $k\times k$ matrices ($k<n$), and $H_1$,$H_2$ ...
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108 views

Determine the dimension of the subspace of all matrices that commutes with an all 1 matrix and find a basis

Let $\Gamma$ denote the $n$-by-$n$ matrix which has $1$ in all its entries. Denote $$S := \{A \in \mathcal{M}_n(\mathbb{R}) : A\Gamma = \Gamma A\}$$ Determine the dimension of $S$, and find a basis ...
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55 views

projection as a linear transformation

I completely understand how projection matrix formula: $$P = A(A^TA)^{-1}A^T$$is derived from: $$ A^T(b - A\hat{x} ) = 0$$ but what I don't understand is the "story proof" or the "...
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78 views

How to derive Lagrangian associated with Projected Variance Maximisation approach to PCA with orthogonal projection

In Projected Variance Maximisation approach to PCA, given orthogonal projection $P_U$ of unlabelled data into the space spanned by the columns of U, s.t projection of the data point $x^{(i)}$ is given ...
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1answer
95 views

Null-Space Projection with a Long Sparse Matrix

I want to solve the classical projection problem: given $x_0:n\times 1$ and a sparse matrix $C: m \times n$, solve for: $$ x = \operatorname{argmin}|x-x_0|^2,\ s.t. \\ Cx = 0 $$ The three usual ways ...
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1answer
98 views

Intersection of ranges of projections

Let $A$ be a unital $\mathrm{C}^*$-algebra and $p,\,q$ non-zero projections in $A$. Consider two faithful representations of $A$ on Hilbert spaces $\mathsf{H}_1,\,\mathsf{H}_2$. Is it possible for the ...
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10 views

Is there any relationship between the degeneracy of a matrix, and the degeneracy of its compression?

Let $A$ be a symmetric $n\times n$ matrix. The $m\times m$ matrix $B$, where $m \leq n$, is called a compression of $A$ if there exists an orthogonal projection $P$ onto a subspace of dimension $m$ ...
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28 views

How to show that a projection matrix is invariant w.r.t the choice of a g-inverse

Let X be an $(nxk)$-matrix (not necessarily of full rank) and $P = X(X'X)^-X'$, i should show that P is invariant with respect to the choice of the g-inverse of $X'X$. As a hint I should first verify ...
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74 views

Consequences of this powerful result?

Here is a statement about "complete system of projectors" : Let $E$ a $\mathbb{K}-$vector space and $p_1,...p_n$, $n$ linear maps. For all $i,j\in\{1,...,n\}$ : $p_i \circ p_j=0$ with $i\...
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6 views

Projection of 2D images on 3D surfaces preserving the distances

I am having the problem of projecting 2D images on 3D surfaces. If we use the orthographic projection, we can obtain the 2D image projected on the 3D surface along a certain direction but without ...
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32 views

Projection and Reflection matrices in different bases

Given a plane $\pi=span(v,w)$ where $v=(3,-2,6)$ and $w=(3,5,-8)$. We have constructed orthonormal basis $B=(u_1,u_2,u_3)$ such that $\pi=span(u_1,u_2)$ and $u_3$ is normal to the plane. Using the ...
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13 views

Eigenvectors after orthogonal projection

For my problem I need to compute the eigenvalues of a matrix after projecting it on a lower dimensional sub space. The challenge is that I have to do this for many different projections. Hence, speed ...
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21 views

Doubt in primary decomposition theorem from Hoffman and Kunze.

The part of the theorem which I am not able to understand is that if $p={p_1}^{r_1}....{p_k}^{r_k}$ is the minimal polynomial of $T$.Then if $T_i$ is the operator induced on $W_i$ by $T$ then the ...
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30 views

A function of a matrix is a projection operator

Let $P_1, P_2, \cdots, P_k$ be projection operators satisfying $\sum_{i=1}^{k} P_i = I$. The function $\Phi(X)$ is defined as follows: $$ \Phi(X) = \sum_{i=1}^{k} P_i X P_i$$ Show that $\Phi^2 = \Phi$....
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7 views

Kronecker product of two projection matrices

Let $M$ and $N$ denote subspaces of a finite-dimensional Euclidian space such that $N \subset M$. Let $P_M$ and $P_N$ be the projection matrix onto the subspaces $M$ and $N$ respectively. We know that ...
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79 views

Is this projection matrix?

What we know about projection matrices: $A^T=A $ $A=A^2 $ I want to prove that $(I-A)^T $is also projection matrix : My solution: $(I-A)^T =I-A^T=I-A=I-A^2$ we can see that $ I-A=I-A^2 => 0=A-A^2$, ...
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51 views

Why is projection matrix diagonalisable?

Suppose $π:V\to V$ is a projection matrix, does it follows that its eigenvalues are $0, 1$? Is $π$ diagonalisable? One of the following answers is true and the other is false, but they both seem true ...
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156 views

Find the matrix of the linear transformation $T : \Bbb R^3 \to \Bbb R^3$ that projects a vector onto the plane $x − y + 2z = 0$

Find the matrix of the linear transformation $T : \Bbb R^3 \to \Bbb R^3$ that projects a vector onto the plane $x − y + 2z = 0$. I know how to find matrix of linear transformation when projecting ...
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31 views

Stabiliser a Conic

I am trying to solve: Consider the conic $C = Z(X_{0}X_{1} - X_{2}^{2})$ in the projective plane. (a) Find the pointwise stabiliser of $C$ in $PGL(3,K)$ (b) Find the setwise stabiliser of $C$ in $PGL(...
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34 views

Given 2 matrices, find T

Trying to find $T$ given $C=T^{-1}BT$ $B = A(A^TA)^{-1}A^T \in \mathbb{R}^{m \times m}$ given $A=\begin{bmatrix}A_1 \\ A_2\end{bmatrix}$ with $A_1 \in \mathbb{R}^{n \times n} $ is invertible and $A_2 \...
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36 views

remove direction of non-standard basis from matrix

Given: G1. C is $n\times n$ positive semidefinite matrix G2. For positive semidefinite matrix SVD decomposition is equal to eigen value decomposition. G3. Singular Value Decomposition of C $SVD(C) = U,...
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165 views

Prove $AA^+$ projects onto the column space of $A$

Prove that $AA^+$ is the projection operator onto the column space of $A$ If $A$ has independent column vectors $A^TA$ is invertible and the projection operator onto the column space of $A$ is $P=A(A^...
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58 views

Range of Idempotent matrices

Suppose $P_1$ and $P_2$ are $n \times n$ matrices satisfying $P_1^2 = P_1$, $P_2^2 = P_2$, and $P_1 P_2 = P_2 P_1$. Prove that $\operatorname{range}(P_1P_2) =\operatorname{range}P_1\cap\operatorname{...
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29 views

Position vector vs normal vector

I am a bit lost here trying to understand "position vector at the origin parallel to the normal" This is related to a projection of a plane onto another plane, that has the normal[1, 2, 3] ...
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1answer
51 views

Matrix expressions for the oblique projection onto subspace L in the direction of subspace K

In the past, I have had to write 3D visualization programs where, in a natural way, oblique projections onto a plane where needed. Each time, I had to develop a specific routine. Later on, I ...
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9 views

Align axis of a tool to world coordinate system

I have a world co-ordinate system. (Let's say x_1, y_1, z_1) My tool moves in this space and has some co-ordinate (x, y, z) and rotation (alpha, beta, gamma) respectively in x, y and z. Using lasers I ...
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27 views

Projection matrix multiplied by projection matrix.

Let $X_1,X_2,...X_n$ be $n$ x $1$ vectors. Let $X_j$ be the matrix composed by the vectors $X_1,...,X_j$ and $X_i$ be the matrix composed by the vectors $X_1,...,X_i$, where $i<j$. The projection ...
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38 views

Representing projection operator in terms of orthonormal basis

The question given is : Let $P: V \rightarrow V$ be a projection operator, i.e., $P^{2}=P .$ If $V$ is finite dimensional, then show that $\operatorname{tr}(P)$ is the dimension of the subspace being ...
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15 views

Column space of matrix X

How to show that C(X)=C(XA) when A is a p by p nonsingular matrix? X is a n by p full rank matrix and we can decompose it into (X1,X2)
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16 views

Pre and post multiply by a projection matrix

What happens to a given matrix A, when you pre and post multiply it by a orthogonal projection matrix?
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32 views

prove that covariance matrix can be expressed as $XCX^\top$ with C is centering matrix

How to prove $\sum_{i=1}^{n}{(x_i-\bar{x}_n)(x_i-\bar{x}_n)^2}$ can be expressed as $XCX^\top$ with $C = I_n -\frac{1}{n} 11^\top $? I read this answer but it did not lead me anywhere because I ...
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21 views

Clarification on terminology: the name for reducing (projecting?) a rank deficient matrix equation to a full rank equation of lower dimension

Context For a given integer $N$, let $V$ be the vector space $\mathbb{C}^N$. Consider the linear equation $$\mathbf{A} \mathbf{x} = \mathbf{b}$$ where, $\mathbf{x},\mathbf{b} \in \mathbb{C}^N$ and $\...
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25 views

What is the relationship between the spectrum of a matrix, and the spectrum after it is projected?

I have a matrix, $Z$, which happens to be diagonal. I am projecting it onto a lower-dimensional subspace, using a set of vectors, $\{y_i\}$. I write this as, $$\xi_{ij} = \langle y_i| Z |y_j\rangle$$ ...
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90 views

How can I make an idempotent projection to an Hermitian operator in a Hilbert space?

I have tried to solve the following question: Consider the Hilbert space $\mathbb R^3$. Find the projection to the 2-dimensional subspace $$\text{span}\{|0\rangle+|1\rangle,|0\rangle+|2\rangle\}$$. I ...

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