# Questions tagged [orthogonal-matrices]

An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.

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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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### 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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### 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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### 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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### 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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### 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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### 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 $d_C(\sigma,\mu)=1$,...
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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 ...
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 $... 1 vote 5 answers 574 views ### Solution for$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 ... 0 votes 1 answer 115 views ### 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= \... 2 votes 1 answer 93 views ### Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here. Let$m < n \in \mathbb{N}$. Let us equip$\mathbb{R}^m, \mathbb{R}^n $with their ... 4 votes 1 answer 697 views ### LU decomposition for orthogonal or unitary matrices? Is there any references on LU decomposition for orthogonal or unitary matrices? It seems to me that the diagonal entries of$U$has some nice structure regarding to the Euler angles of the original ... 24 votes 2 answers 830 views ### Simple conjecture about rational orthogonal matrices and lattices The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ... 1 vote 0 answers 758 views ### How to project a matrix to a unitary matrix? Given a nonzero vector$v \in \mathbb{R}^n$, we all know that it's projection onto the unit$\ell_2$ball is just$\frac{v}{\|v\|}$. Let$X$be some nonzero$n \times n$matrix. What is the projection ... 2 votes 1 answer 438 views ### Minimize matrix norm over the unitary matrices Suppose$C_1$and$C_2$are some fixed$n \times n$matrices. Define the norm$\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$. What is$\min_U \|C_1 U C_2 \|$? Here$U$ranges over the$n \times n$unitary ... 0 votes 0 answers 32 views ### Condition on the point cloud matrix making the points "generic" in the uniform sense For a matrix$X\in\mathbb{R}^{d\times n}$, what condition can I impose on$X$to make the collection of its columns generic in the sense that they look like the result of uniformly sampling a convex ... 6 votes 1 answer 249 views ### How can I solve an orthogonal-constrained Sylvester equation? I am currently facing a Sylvester equation$AX+XB = C$where$A$,$B$,$C$are all symmetric and a special constraint here is that$X$should be orthogonal. The Sylvester equation itself may not ... 4 votes 1 answer 221 views ### Parametrizing quotient of matrices by the orthogonal group I am trying to parametrize the collection of$d\times m$real matrices quotient$d\times d$orthogonal matrices. Formally, define$\sim$on$\mathbb{R}^{d\times m}$by$X\sim Y$if there exists an ... 6 votes 1 answer 325 views ### Can we choose smoothly the singular vectors of a matrix?$\newcommand{\GLm}{\text{GL}_n^-}$Let$A$be a real$n \times n$matrix with non-positive determinant. Suppose that the smallest singular value of$A$is strictly smaller than all the others (it has ... 3 votes 1 answer 442 views ### Upper bound on the sectional curvature of the orthogonal group Consider the orthogonal group$O(n)$as a Riemannian manifold endowed with the usual (bi-invariant) metric$\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$for tangent vectors$P, Q$, with$$T_A O(n) =... 4 votes 1 answer 260 views ### 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(...
Context: I am trying to show the convergence of an optimization method which includes orthogonalization in the update step. Problem: Let's say I have real matrices $A, B \in \mathcal{R}^{n xm}$. If ...