Questions tagged [orthogonal-matrices]
An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.
115 questions
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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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3
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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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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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) =...
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Bounding the Frobenius norm of orthogonalised matrices
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 ...
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Real orthogonal and sign [closed]
I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector.
It should be straight in ...
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Orthogonal similarity of adjacency matrices of graphs which are cospectral and have a common equitable partition
Let $G$ and $H$ be two undirected graphs of the same order (i.e., they have the same number of vertices). Denote by $A_G$ and $A_H$ the corresponding adjacency matrices. Furthermore, denote by $\bar G$...
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Parametrising a sparse orthogonal matrix
I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $AA^𝑇...
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Is there a standard name for (non-square) matrices with orthonormal columns?
One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name ...
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Generalization of Jordan's Lemma $A^2=B^2=I$ can be 2-block diagonalized
One of Jordan's lemma states that if two orthogonal matrices $A,B$ are such that $A^2=B^2=I$, then they can be co-diagonalized by block of size 2.
(the proof is easy, consider $x$ an eigenvector of $A+...
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Is there a "formula" for the point in $\text{SO}(n)$ which is closest to a given matrix?
$\newcommand{\Sig}{\Sigma}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\distSO}[1]{\dist(#1,\SO)}$
$\newcommand{\distO}[1]{\text{dist}(#1,\On)}$
$\newcommand{\tildistSO}[1]{\operatorname{...
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Infinitesimal matrix rotation towards orthogonality
TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...
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A question about special linear group
Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying
$$
AGA^{-1} \in SO(n,\mathbb R)?
$$
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Geodesics on SO(3)
I have two 3D rotations about the origin, represented as
$3 \times 3$ orthogonal matrices $M_1$ and $M_2$
(specified by numerical entries),
and I would like to interpolate (and compute)
a continuous ...
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When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?
Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding ...
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Orthogonal Matrices and Cosets (translates) of Linear Subspaces
Let $M_n(F_2)$ be the vector space of all $n\times n$ matrices over the finite field $F_2$. Let $O(n)\subset M_n(F_2)$ be the set of all orthogonal matrices and $W\subseteq O(n)$ be an affine subspace ...
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Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...
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Inverse of matrix $D + ADA^T$
Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently?
$$(D + ADA^T)^{-1}$$
Hints or references are ...
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Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the ...
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Finding a similarities and differences of sent of matrices
Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,
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Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix
Given two $N \times N$ symmetric matrices $A, B$, where $A$ is positive semidefinite while $B$ is not positive semidefinite. I am interested in solving unitary constrained trace maximization problem:
...
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Characterizing a subclass of row-orthogonal matrices
Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...
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Positive definite matrices diagonalised by orthogonal matrices that are also involutions
Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$.
I want to know when this matrix is also an involution, i.e., $P^2 = I$.
If there is any ...
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A question about permutation matrices
This question is trying to abstract out in a self-contained way the point that is probably being made in page 6 of this paper, https://arxiv.org/pdf/1604.03544.pdf and why Theorem 4.1 there works.
...
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What's the best orthonormal matrix to align two matrices in the operator norm sense?
Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that
\begin{equation}
UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F},
\end{equation}
where $USV^\top$ is ...
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maximum of orthogonal vectors
$$v_1=(x_1,x_2,x_3\cdot\cdot\cdot,x_n)$$is such a vector. By changing its signs and positions of each component $x_i$, we can get different vectors. When n is odd, it's impossible for any of ...
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Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?
Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...
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What is the term for a matrix whose columns are orthogonal?
What is the term for a matrix whose columns are mutually orthogonal, but not necessarily othonormal?
I can't name such a matrix "orthogonal" because that would imply that all columns are unit vectors....
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What is the term for this type of matrix?
Is there an established term for the following type of square matrices?
$\begin{pmatrix}
c & c & c & c & \cdots & c & c \\
c & a & b & b & \cdots & b & ...
15
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matrix inequality with orthogonal matrices
I would like to know if for $A,B\in SO(3)$ the inequality
$$
\|AB-BA\|_F\leq \|A-I\|_F\|B-I\|_F
$$
holds, where $\|\cdot\|_F$ denotes the Frobenius norm and $I$ the identity matrix. Using the identity
...
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Bounding the non-multiplicativity of isometric projection
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition:
$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$.
In particular the orthogonal factor is given by $$O_A=A(\...
CommunityBot
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Orthogonal polynomials of the second kind
Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...
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What is special in dimension $2$ (When characterizing isometries using the cofactor matrix)?
Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \...
CommunityBot
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When does isometric projection respect multiplication?
Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$,
( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition).
...
CommunityBot
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1
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A question on orthogonal matrix [closed]
Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.
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