Questions tagged [orthogonal-matrices]
Matrices with orthonormalized rows and columns. An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose. For complex matrices the analogous term is *unitary*, meaning the inverse is equal to its conjugate transpose.
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Can a Householder reflector (transform) be extended to a product of vectors?
Can a Householder reflector (transform) be extended to a product of vectors? Specifically, say we have two routines:
$$\begin{align}
h_m(i,x) =& (\alpha,v)\textrm{ where } x-\alpha\langle x,v\...
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How to solve second-order objective function with orthogonal constraint
I want to minimize $tr(P^TAP-P^TB)$ with the constraint $P^TP=I$.
$P, A, B$ are all matrices, $I$ is the identity matrix, $tr$ means the trace of a matrix.
P: size(m,l)
A: size(m,m) non-symmetric ...
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l2-norm of matrix products
I have a matrix $Q\in\mathbb{R}^{m\times n}(m>n)$, where the columns of Q are unit and mutually orthogonal. $W\in \mathbb{R}^{m\times m}$ is a diagonal matrix with diagonal elements 0 or 1, and $WQ$...
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Decompose a $3 \times 3$ orthogonal matrix into a product of rotation and reflection matrices
It holds that every orthogonal matrix can be expressed as a product of rotation and reflection matrices.
We can prove that every $2 \times 2$ orthogonal matrix can be represented in the form
$$\begin{...
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Calculate differential $dF_j$ of $F:j\mapsto j^2+I$ on $O(n)$
Given the map $F:j\mapsto j^2+I$ on the orthogonal group $O(n)$, what is the differential $dF_j$ ? How do I calculate this?
I am trying to understand Example 7 in Lecture Notes on Symmetric Spaces by ...
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Hadamard matrix of order 28 via Paley 1 construction [closed]
The Paley 1 construction method allows to construct an Hadamard matrix (square matrices with orthogonal columns and entries equal to $1$ or $-1$ ) of order N = q+1 where q = 3 (mod 4) is a prime power:...
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Explicit example of a set of coset representatives of $U(n)$ within $O(2n)$
I understand how to identify a unitary group $U(n)$ with the elements of the orthogonal group $O(2n)$ which commute with a linear complex structure $J$. I am also aware of the "two-out-of-three&...
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Conjugacy action of $SO(2m)$ on $O(2m)/U(m)$
I seek intuition about the symmetric space $S$, the set of orthogonal complex structures in $\mathbb{R}^n$ for even $n=2m$.
I am finding J.H.Eschenburg's Lecture Notes on Symmetric Spaces very helpful....
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Closed form formula for $(\mathbb I+R)^{-1}$ where $R$ is an orthogonal matrix?
Is there a closed form formula for $(\mathbb I+R)^{-1}$, where $R$ is an orthogonal matrix?
There are formulas for inverse of matrix sums, but I couldn't find one that evaluated to a simple result.
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map from spin to special orthogonal in Magma [closed]
Let $G:=\operatorname{Spin}(7,5)$. How to construct in Magma the map $G \rightarrow G/Z(G) $ where $Z(G)$ is the center. I get this from Magma:
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How to think of regular orbits of a finite orthogonal group?
Let $G$ be a finite subgroup of $O(n)$ acting on $\mathbb R^n$. A regular orbit of the action is one such that the cardinality of the orbit is equal to $|G|$.
I am at a loss as to how to prove some ...
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Given $A_i, B_i \in \mathbb{R}^{k \times d}$, minimize $\sum_{i} \lVert U A_i V^T - B_i \rVert_F^2 $ over orthogonal $U, V$.
Given a collection of rectangular matrices $A_i, B_i \in \mathbb{R}^{k \times d}$ for $1 \leq i \leq n$, I am looking for an analytical solution for orthogonal matrices $U \in \mathbb{R}^{k \times k}$ ...
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Why the orthogonal projection give minimal?
Given the data $ \vec{x}=(-2,-1,1,2) )$ and $( y=(1,1,-1,1) )$. Use an orthogonal projection to determine the coefficients $( a_{0}, a_{1}, a_{2} )$ of the quadratic polynomial function
$
\begin{...
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Proof rotation matrix is symmetric when Trace is -1
For a rotation matrix on SO(3), IE 3 dimensional, if the trace is -1 how do you prove it is symmetric?
Intuitively it makes sense as this is 180 degree rotation but I don't see an obvious proof.
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Interpolation in $O(p,q,r,\mathbb{R})$
Definite Setting: $SO(n,\mathbb{R})$ vs $O(n,\mathbb{R})$
If I have a rotation matrix $R_0\in SO(n,\mathbb{R})$ and a rotation matrix $R_1 \in SO(n,\mathbb{R})$ I can interpolate between the two by ...
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