# Questions tagged [orthogonal-matrices]

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

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### Cardinal of finite orthogonal groups

Let $p \neq 2$ and let $F$ be a $p$-adic field with ring of integers $\mathcal{O}$ and maximal ideal $\mathfrak{p}$. By a quadratic space $V_{\mathcal{O}}$ of dimension $d$ over $\mathcal{O}$, I mean ...
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### Inequality involving minors of an orthogonal matrix

Fix $n \geq 3$ and take any orthonormal vectors $x,y,z \in \mathbb{R}^n$. Let also $A \in M_n(\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following ...
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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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### 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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### 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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### 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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### 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 ...
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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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### 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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