Questions tagged [projection-matrices]

Question 1

Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that $$ \underset{P^2 = P = P^T,\; \text{rank}(P)...

Question 2

I am interested in the following matrix Here $A$ and $B$ are complex numbers. Are matrices of this type known in the literature? Is there some "obvious" reason why they should have maximal ...

Question 3

Let $P$ be a projection in an n-dimensional vector space $V$. Let $\{z_{1}, ..., z_{r}\}$ be a basis for range($P$). Now extend it to a basis for $V$, i.e., $\{z_{1}, ..., z_{r}, z_{r+1}, ..., z_{n}\}$...

Question 4

Given a projection matrix $P$ (not necessarily an orthogonal projection), and another matrix $A$ where $\mathrm{eig}(A)$ all have magnitude strictly less than $1$, and both $P,A$ are real-valued. Can ...

Question 5

Please prove or disprove the following conjecture. Conjecture. Let $B \in \mathbb{R}^{n \times n}$ be a column stochastic (also known as left stochastic) matrix, and let $ \pi \in \mathbb{R}^n$ be ...

Question 6

Let $E: ax+by+cz=0$ be a plane in ${\Bbb R}^3$. Hence, its normal vector is $\vec{n} = ({a,b,c})$. To find the orthogonal projection matrix that maps every point in ${\Bbb R}^3$ onto $E$, we may write ...

Question 7

I am aware that, given a matrix $A$, the orthogonal projection $P$ onto $\operatorname{Range}(A)$ is given by, $$ P = A \left( A^T A \right)^{-1} A^T \tag{1} $$ Is the converse true? That is, if $P$ ...

Question 8

Let $A \in \mathbb{R}^{n\times n}$ be a symmetric matrix $A = A'$. The eigenvalues $\lambda$ and associated eigenvectors $x$ of this matrix are defined as usual as the solutions of $Ax = \lambda x$. ...

Question 9

In Gaussian splatting, when projecting a 3D Gaussian distribution onto a 2D image plane, the transformed 2D covariance matrix ${\bf \Sigma}'$ is given by: $${\bf \Sigma}' = {\bf J} {\bf W} {\bf \Sigma}...

Question 10

I am working on the following problem from chapter 7 of Sheldon Axler's Linear Algebra Done Right (4th edition): F14. Suppose $U$ and $W$ are subspaces of $V$ such that $\| P_U − P_W \| < 1$. ...

Question 11

Let $J_m$ be an $m \times m$ matrix whose entries are all ones and $K_n$ the adjacency matrix of the complete graph of $n$ vertices. Consider the Kronecker product $$A = J_m \otimes K_{n_1} \otimes K_{...

Question 12

Suppose we have a real $m \times n$ matrix $X$ and an orthogonal projection $\Pi$ onto $\mathbb{R}^m$. I am trying to prove or disprove the following statement $$ \|X\|_* = \| \Pi X \|_* \iff X = \Pi ...

Question 13

Let $A \in \mathbb{R}^{N\times N}$ symmetric matrix with eigenvalues $0 \leq \lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_N \leq 2$ associated to eigenvectors $u_1, \ldots, u_N$. We want to ...

Question 14

Starting from some positive semi-definite $M^{(0)}$, does the sequence $$ M^{(r+1)} = A U^{(r)} V^{(r)\mathsf{H}} B $$ converge where $A, B$ are positive semi-definite and $U^{(r)}, V^{(r)}$ are left ...

Question 15

Let $A\in\mathbb R^{d\times d}$ be a symmetric matrix whose SVD is $A = U \Lambda U^T$, where $U\in\mathbb R^{d\times d}$ is orthogonal and $\Lambda\in\mathbb R^{d\times d}$ is diagonal. Let $B\in\...

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Questions tagged [projection-matrices]

Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?

Matrix with monomial entries

range and null space of a projection

Eigenvalues of $PA$ with $P$ a projection and $A$ having eigenvalues inside open unit disk

Norm inequality for a column stochastic matrix whose columns are projected onto a subspace orthogonal to the Perron eigenvector

Matrix for orthogonal projection onto a plane

Orthogonal projection matrices

Properties of the matrices generated by the eigenvectors of a symmetric matrix

Why is the linear approximation term omitted in the 3D-to-2D covariance projection formula for Gaussian splatting?

Suppose $U$ and $W$ are subspaces of $V$ such that $\| P_U − P_W \| < 1$. Prove that $\dim U = \dim W$

Orthogonal projection matrix of a Kronecker product of matrices

Nuclear norm (trace norm) after applying an orthogonal projection

Norm of the difference of projection operators

Convergence of a sequence constructed iteratively from its orthogonal projection

SVD of the projection of a matrix

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