Questions tagged [orthogonal-matrices]
An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.
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Computing Haar measure of matrices sampled from SO(n)
I am looking to sample uniform matrices from SO(n).
I know that uniform matrices can be sampled from O(n) by taking the QR decomposition of Gaussian random square matrices and adjusting the sign of ...
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Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)
I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$.
$$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
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Minimizing quadratic objective under orthogonality constraints
The following problem is motivated from Generalized Procrustes Analysis. I am wondering if it is possible to obtain a closed form minimizer (which may involve SVD or some other decomposition of a ...
CommunityBot
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Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...
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Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
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How can one orthogonalize the pointwise sum of two orthogonal sets?
Let $n = 2k$, and suppose that $V = \{v_1, \cdots, v_k\}$ is an orthogonal set in $\mathbb{R}^n$. In other words, the vectors in set $V$ are pairwise orthogonal to each other.
Now, consider a new set $...
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Smallest Singular Value of submatrices of a column-orthogonal matrix
Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\...
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Measuring the "distance" of a matrix from a diagonal matrix
Let $A$ be a $N \times N$ symmetric positive semi-definite matrix with $N \geq 2$. Let $D$ be a diagonal matrix of dimension $N$. I would like to measure how much $A$ "is far" from $D$, i.e. ...
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Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
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One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
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Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$
$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...
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Orthonormal matrices with columns that switch signs
Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w_i|^T |w_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w_i \in \mathbb{R}^{2n}$ is ...
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Construct a permutation matrix from some eigenvectors and eigenvalues
Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...
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Conjecture on the existence of centrosymmetric Hadamard matrices
I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices.
Definition: An $n \times m$ matrix $A = (a_{i,j})$ is ...
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Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
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A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$
Let $p$ be an odd prime, and consider the group
$$\{U\in \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}) : U^{t}U=I \bmod p \}\subseteq \operatorname{SL}_n(\mathbb{Z}/p\mathbb{Z}).$$
I wonder what is the ...
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How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?
Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix:
$$
\min_{s\in\...
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Variant of Wahba's problem
Wahba's problem is the following:
$$\min_R \sum_{k=1}^K \|v_k - Rw_k\|^2$$
where $v_k$ and $w_k$ are arbitrary $3\times 1$ vectors, and $R$ is a rotation matrix (i.e., orthogonal with $\det(R)=1$).
A ...
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Orthonormal solution of overdetermined linear equations
I have a two matrices $A$ and $B$ in $\mathbb{R}^{m \times n }$ ($m \gg $ n) such that there exists an orthonormal matrix $X \in \mathbb{R}^{n \times n }$, such that:
$$AX = B$$
Given that $X$ is ...
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How many householder matrices do I need for constructing a given unitary matrix?
As we know we can construct unitary matrix as $H=H_1H_2\dots,H_k$, by stacking householder matrices $H_i\in \mathbb{R}^{d\times d}$. The number of householder matrices we use, i.e., $k$, determines ...
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Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
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Proving that a product of reflections and an orthogonal matrix is in $\mathrm{SO}_*(V)$
Let $V=(V,b)$ be a finite-dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)...
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Existence of parametrizations of rational orthogonal matrices
I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this?
Thanks....
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What do you call a scaled orthogonal map?
What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, ...
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Decomposition of a 4D rotation into a particular sequence of simple rotations
I asked this question in math.stackexchange two days ago, but no one has answered yet. I suspect it is "hard enough" that it is appropriate to post it here as well. I am new to stackexchage, ...
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Distribution of Submatrix of Orthogonal Matrix
Let $O$ be a matrix sampled from the Haar measure on $O(n)$. Let $X$ be the upperleft $k\times k$ submatrix of $O$.
In a physics research project I am interested in the distribution of $X$, say $\rho(...
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Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
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$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns
I am interested in answering the following question:
Question
For a given $n$, does there exist a $2n \times 2n$ matrix with entries in $\{1, 0, -1\}$ having orthogonal rows and columns with exactly $...
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Reference request: maximal determinant of matrices with pairwise orthogonal rows and entries in $\{1, 0, -1\}$
We know that "Hadamard maximal determinant problem" concerns the largest determinant of a matrix of oder $n$ with entries in $\{-1,1\}$ or $\{0, 1\}$. For $n=4k$, it is the Hadamard ...
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Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
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Orthogonal invariance of (weighted) Laplacian
It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.
Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian
$$...
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Is the group law for SO(2n, R) encoded in so(2n,R)?
Note that this is a partial duplicate of my math.stackexchange question here. In this post I am asking something slightly broader. Note that I am a mathematical physicist and not a representation ...
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Subgroups of $\mathrm{SO}(A_0, \mathbb{F}_p)$
Let $n \geq 3$. Let $A_0$ denote the $n \times n$ symmetric matrix with $1$'s on the antidiagonal and $0$'s everywhere else. We can define the associated special orthogonal group
$$ \mathrm{SO}(A_0, \...
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Maximize the product of Hadamard matrix and a vector
Let $X$ be an $n \times n$ Hadamard matrix (i.e. entries are in $\{-1,1\}$ and rows are orthogonal). For my application, we can assume $n=2^k$.
Given a vector $\bf{w} \in R^n$, I want to find the $X^*$...
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Proving a lemma for a decomposition of orthogonal matrices
Setting
Consider two independent orthogonal matrices, which are decomposed into 4 blocks:
\begin{align}
Q_{1}
=
\left[\begin{array}{cc}
A_{1} & B_{1}\\
C_{1} & D_{1}
\end{array}\right]
,
\,Q_{...
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Matrix equation and spherical harmonics
I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...
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functions with orthogonal Jacobian
I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...
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Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on $O(n)$?
Denote by $d_C(\sigma,\mu)$ the minimal number of transpositions needed to go from a permutation $\sigma$ to a permutation $\mu$. E.g. if $d_C(\sigma,\mu)=0$, then $\sigma=\mu$, if $d_C(\sigma,\mu)=1$,...
CommunityBot
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The invertible matrices $S$ that satisfy $A=SDS^T$
Any real symmetric matrix $A$ can be written as $A=SDS^T$ for some diagonal matrix $D$ and invertible matrix $S$. Let's fix $D$ to be the (diagonal) inertia matrix of $A$, which has an entry $1, -1, 0$...
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Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$
I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have
$$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...
CommunityBot
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The space of positive definite orthogonal matrices
The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite.
$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal.
and the ...
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Minimize smooth function $(x,y) \to f(x,y)$ subject to $x \perp y$
Let $V$ be a finite-dimensional real vector space (e.g space of $m \times n$ real matrices equiped with Hilbert-Schmidt inner product $(A,B) \to \mathrm{tr}(AB^\top)$, and let $f:V^2 \to \mathbb R$, $(...
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Solution for $Xa + X^Tb = c$ where $X^TX = I$? [closed]
There are three known $n\times1$ vectors: $a, b, c$, along with one unknown $n\times n$ matrix: $X$. I am only interested in the $n={2,3}$ cases.
$X$ is $2\times 2$ or $3\times 3$ rotation matrix ...
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Probability of $\ell_1$-norms of vertices of the rotated Hamming cube
Let $O$ be a $d$-dimensional rotation matrix (i.e., it has real entries and $OO^T = O^TO = I$). Let $\mathbf{x}$ be a uniformly random bitstring of length $d$, i.e., $\mathbf{x} \sim U(\{0,1\}^d)$. In ...
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Parametrizing quotient of matrices by the orthogonal group
I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an ...
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The only rotation fields satisfying this PDE are constant
$\newcommand{\div}{\operatorname{div}}$$\newcommand{\SO}{\operatorname{SO(2)}}$$\newcommand{\R}{\operatorname{\mathbb{R}}}$$\newcommand{\bdx}{\partial_x}$$\newcommand{\bdy}{\partial_y}$$\newcommand{\...
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Principal orbit and the generic stabilizer of SO(2n)xSO(2n)
Let $SO(2n)$ denote the special orthogonal group of $2n\times 2n$ matrices over the complex numbers.
Consider the action of $SO(2n)\times SO(2n)$ on the set of $2n\times 2n$ matrices : $ADB^{T}$, ...
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Number of 5x5 matrix permutations without repetitions in rows or columns
Context
In the boardgame Azul, your goal is to complete as much as possible of a $5\times5$ board by placing 25 tiles of 5 different colours (5 tiles of each colour) so that no colour appears twice in ...
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Can an orthogonal matrix move monotonically toward a signed permutation matrix?
The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
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Maximize function on rotation matrices [closed]
Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...
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