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.
1,180 questions
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If the sum of each row and column of an orthogonal matrix equals 1, is it a permutation matrix?
A permutation matrix has a single $1$ in each row and column, with the rest of the entries in that row and column equal to zero.
Question: If I have an $n\times n$ orthogonal matrix $\mathbf{Q}$ ...
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Distance Between Subspaces
Let $P_1$ denote the orthogonal projection onto the subspace $S_1 \subset \mathbb{R}^n$, and correspondingly, we take $P_2$ to be the orthogonal projection onto the subspace $S_2 \subset \mathbb{R}^n$....
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Proof (or reference) that the measure of a set $S\subset\mathbb{R}^n$ is preserved by an orthogonal transformation $V$?
Suppose $S\subset \mathbb{R}^n$ is measurable and has $\text{Vol}(S)$. If $M$ is a linear transformation then it is known that $\text{Vol}(MS)$ is equal to $\text{Vol}(S)$ multiplied by the product of ...
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Can $e^A$ be visualized as a rotation in $\mathbb{R}^3$ when $A$ is a $3 \times 3$ real skew-symmetric matrix?
I know that for any skew-symmetric matrix $A$, the exponential $e^A$ is an orthogonal matrix due to the identity:
$(e^A)^T = e^{-A} = (e^A)^{-1}$
However, I’m looking for a deeper geometric intuition:
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Find orthogonal $O$ such that $OMO'$ has constant diagonal
Let $M$ be a given symmetric real matrix.
Can we always find an orthogonal matrix $O$ such that
$$OMO^\top$$
has constant diagonal? That is, $(OMO^\top)_{ii}=d$ for some constant $d$ independent of $I$...
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$SO(d)$ invariant functions are $O(d)$ invariant for narrow matrix multiplication
Background
Let $ d, n \geq 2$ be natural numbers. Consider the groups
$$
O(d)=\left\{R \in \mathbb{R}^{d \times d} \mid \quad R R^{T}=I_{d}\right\}, \quad S O(d)=\{R \in O(d) \text { and } \...
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How to make sure that an orthogonal transformation results in only non-negative off-diagonal elements?
This is a very specific question but I have no idea on where to start. Let's say I have an orthogonal transformation $A = VB V^T$, where $B$ is a diagonal matrix and $V$ is an orthogonal matrix. For a ...
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number of integer vector near a hyperplane.
Let $x \in \mathbb{Z}^N$ be a non-zero integer vector. I am interested in obtaining an upper bound for the following quantity
$$ \# \left\{ y \in \mathbb{Z}^N: \|y\| < Y, \operatorname{dist} \left(...
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The order 4 isometry of the tetrahedron
Consider a tetrahedron centered at the origin. For example with the vertices $ (\pm1,0,-1/\sqrt{2}),(0,\pm1,1/\sqrt{2})$. The isometry group of the tetrahedron is $ S_4 $ and the subgroup of ...
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Why does the orthogonal matrix act from the right in the Procrustes problem?
I have a problem with understanding the Procrustes optimization task, that is finding matrix $Q$ which transforms vectors stored in matrix $B$ so that the transformed vectors $BQ$ are as close as ...
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What is the smallest continuous group that contains $S(N)$?
We know that $O(N)$, the group of orthogonal matrices of dimension $N$, contains $S(N)$ as a subgroup, the group of permutations of $N$ elements. Another example is $U(N)$, the group of unitary ...
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Which double covers of Lorentzian orthogonal groups restrict to spin groups?
It is known there are in general $8$ irreducible double covers of any Lorentzian orthogonal group $\operatorname{O}(1,q) \cong \operatorname{O}(q,1)$ for $q > 2$. These $8$ covers correspond to the ...
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Eigenvalues of a complex orthogonal matrix
Property 1:A complex orthogonal matrix must have eigenvalues with modulus 1.
Property 2: If all entries in the matrix are real (real orthogonal matrix), then the eigenvalues must be $\pm 1$
Proof of ...
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How to solve $\dot{A}(t)=A(t)B(t)$
I have a matrix differential equation $\dot{A}(t) = A(t)B(t)$ where $A$ is an orthonormal matrix and $B$ is a known skew-symmetric matrix, both 3-by-3. The farthest I could get in finding a general ...
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Properties of a matrix such that $A+A^T= \alpha Id$ [closed]
What can be said about the properties of an $n\times n$ invertible matrix $A$ with real entries such that $A+A^T=\alpha I_n$, where $\alpha\in \mathbb{R}$?
Particularly, I am looking for properties ...