All Questions
Tagged with orthogonal-matrix or orthogonal-matrices
115 questions
4
votes
4
answers
2k
views
I want a smooth orthogonalization process
The following question is related to research I am doing on reinforcement learning on manifolds.
I have a set of basis vectors $\boldsymbol{B} = \{\boldsymbol{b}_1,\dots,\boldsymbol{b}_k\}$ that span ...
- 155
3
votes
1
answer
234
views
Is there a matrix that has the completely opposite effect of a Hadamard matrix?
First, let me provide some background on the problem:
In the field of Large Language Model quantization/compressions, outliers (abs of outliers are much larger than the mean of abs of all elements in ...
- 31
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 27
3
votes
1
answer
187
views
Eigenvalues of certain matrices
We write $R(\theta)=\left(\begin{smallmatrix}\cos(2\pi\theta)&\sin(2\pi\theta)\\ -\sin(2\pi\theta)&\cos(2\pi\theta)\end{smallmatrix}\right)$ for any $\theta\in\mathbb R$.
Let $d,m,n,r$ be a ...
- 2,446
0
votes
0
answers
96
views
When can a point be reconstructed from relative angle measurements?
Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
- 427
2
votes
1
answer
78
views
Reference for irreducible representations of $\mathcal{O}(n)\ni O\mapsto O^{\otimes k}$
This MO answer cites the Goodman-Wallach book to affirm that:
$$\mathrm{Sym}^k\left(\mathbb{R}^n\right)=\mathcal{H}^k\oplus q\mathcal{H}^{k-2}\oplus q^2\mathcal{H}^{k-4}\oplus\cdots$$
with $\mathrm{...
- 167
1
vote
0
answers
43
views
Moments on the Stiefel manifold
Let $S_{n, k} = \{V \in \mathbb{R}^{n \times k} : V^T V = I_k\}$ denote the Stiefel manifold, $1 \leq k \leq n$.
Let $P \in \mathbb{R}^{n \times n}$ denote a symmetric real, positive definite matrix, ...
- 380
7
votes
2
answers
201
views
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
Let $M_n(\mathbb{C})$ represent the space of $n \times n$ matrices over $\mathbb{C}$. We will think of it as a $\mathbb{C}$-vector space.
Notice that if $A \in M_n(\mathbb{C})$ is invertible, then the ...
- 431
1
vote
1
answer
114
views
Sum of squares of $k\times k$ cofactors is $1$ for an orthonormal matrix [closed]
Let $n,k\in \mathbb N$ with $k\leq n$. Let $A$ be an $n\times n$ real orthonormal matrix. Fix any $k$ rows of $A$ and from there consider every possible $k\times k$ cofactors and there will be exactly ...
- 49
1
vote
0
answers
60
views
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 ...
- 81
2
votes
0
answers
80
views
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 ...
- 21
8
votes
2
answers
750
views
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 ...
- 181
0
votes
0
answers
97
views
Can a Tangent Space always be expressed with “more structure” than just a vector space (e.g. a choice of basis for Stiefel manifold)
I'm currently trying to read about the Stiefel manifold, or set of all $p$ orthonormal $n$-dimensional vectors embedded in $\mathbb{R}^{n\times p}$.
$$\mathcal{V}_p(\mathbb{R}^n) = \{U \in \mathbb{R}^{...
- 1
1
vote
1
answer
138
views
Orthogonal vectors translation using standard vectors
When $n=2m$, let us consider the following vectors $\mathbf{v}_1,\ldots, \mathbf{v}_n$ in $\mathbb{R}^n$
$$\mathbf{v}_q=(v_{1q},\ldots,v_{n,q})$$
$$v_{p,q}=\sin\Big(\frac{pq}{n+1}\pi\Big)$$
It is ...
- 4,058
1
vote
1
answer
214
views
Conditions of P for existence of orthogonal matrix Q and permutation matrix U satisfying QP = PU
Question: Let $P\in \mathbb{R}^{d\times n}$ be a $d$-rank real matrix and $PP^T = c I_d$ with a certain constant $c > 0$. Under what additional conditions of $P$ does there exist an orthogonal ...
- 187
14
votes
4
answers
2k
views
Measuring the "distance" of a matrix from a diagonal matrix
Let $A$ be a $N \times N$ symmetric positive semi-definite matrix with $N \geq 2$. Let $D$ be a diagonal matrix of dimension $N$. I would like to measure how much $A$ "is far" from $D$, i.e. ...
- 251
8
votes
2
answers
519
views
Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices
My question is motivated by this one, but within real matrices instead of complex ones.
${\bf Sym}_n(\mathbb R)$ is a vector space of dimension $N=\frac{n(n+1)}2$. Equipped with the scalar product $\...
- 52.3k
2
votes
1
answer
98
views
One question about nega-cyclic Hadamard matrices
Let $n$ be a multiple of $4$, is there any $n \times n$ negacyclic Hadamard matrix? If yes - how to construct it? If no - why?
Here an $n \times n$ nega-cyclic matrix is a square matrix of the form:
\...
- 696
1
vote
1
answer
112
views
Orthonormal matrices with columns that switch signs
Consider an orthonormal matrix $W\in\mathbb{R}^{2n\times 2n}$ that satisfies the "abs property" $$|w_i|^T |w_{i+n}|=1$$ for all $i \in \{1,2,\ldots,n\}$, where $w_i \in \mathbb{R}^{2n}$ is ...
- 13
1
vote
0
answers
237
views
Construct a permutation matrix from some eigenvectors and eigenvalues
Given $n$ orthonormal vectors $v_1, \dots, v_n \in \mathbb R^d$, where $d > n$, we can show that there are many orthogonal matrices $X$ of size $d$ such that $v_1, \dots, v_n$ are eigenvectors of $...
- 111
8
votes
2
answers
440
views
Show $\langle \log(R), \log(R^{-1}S) \rangle \geq \langle \log(R), \log(S) - \log(R) \rangle $ for all $R,S \in \mathrm{SO}(3)$
$\DeclareMathOperator\SO{SO}$I have a similar question to one I asked a few days ago. Lately, I've been researching Lie groups equipped with bi-invariant Riemannian metrics. One common object is $\SO(...
- 518
8
votes
2
answers
642
views
Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?
$\DeclareMathOperator\SO{SO}$I asked this initially in math stack exchange, but thought to ask it here since it is more advanced and related to my research topic. I study optimization on Lie groups ...
- 518
2
votes
1
answer
207
views
A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$
Let $p$ be an odd prime, and consider the group
$$\{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}).$$
I wonder what is the ...
- 253
8
votes
3
answers
1k
views
Conjecture on the existence of centrosymmetric Hadamard matrices
I work with centrosymmetric matrices and recently have started exploring the question of the existence of centrosymmetric Hadamard matrices.
Definition: An $n \times m$ matrix $A = (a_{i,j})$ is ...
- 539
1
vote
3
answers
345
views
How to optimize for the best fit nonuniform-scale-rotation to a given 3×3 matrix?
Given a matrix $L\in \mathbb{R}^{3 \times 3}$, I'm looking for a method to find the closest (in a least squares sense) product of a non-uniform scaling matrix and a rotation matrix:
$$
\min_{s\in\...
- 723
3
votes
1
answer
330
views
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 ...
1
vote
1
answer
158
views
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 ...
- 113
2
votes
2
answers
185
views
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 ...
- 123
3
votes
0
answers
46
views
Evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthonormal matrices of a certain size
I am trying to evaluate $\int_\phi e^{tr(RM)} dR$ where $\phi$ is a set of all real orthogonal matrices of a certain size. $M$ is an arbitrary real matrix (of a certain size).
This is equivalent to
$$\...
- 433
2
votes
0
answers
283
views
Proving that a product of reflections and an orthogonal matrix is in $\mathrm{SO}_*(V)$
Let $V=(V,b)$ be a finite-dimensional vector space equipped with $b$ a symmetric and positive definite bilinear form. And let $\{e_1,\dotsc,e_n\}$ be a orthonormal basis for the subspace $\ker((P_A)^t)...
3
votes
2
answers
569
views
What do you call a scaled orthogonal map?
What do you call a linear map of the form $\alpha X$, where $\alpha\in\Bbb R$ and $X\in\mathrm O(V)$ is an orthogonal map ($V$ being some linear space with inner product)? Are there established names, ...
- 13.6k
2
votes
0
answers
102
views
Decomposition of a 4D rotation into a particular sequence of simple rotations
I asked this question in math.stackexchange two days ago, but no one has answered yet. I suspect it is "hard enough" that it is appropriate to post it here as well. I am new to stackexchage, ...
- 21
2
votes
1
answer
260
views
Non-Fourier complete orthogonal basis?
The Fourier Transform (FT)
Is orthogonal: inner product of one basis, $e^{j\omega_0}$, with any other basis, $e^{j\omega_1}$, is zero
Is invertible: info-preserving, has inverse function
Is energy-...
12
votes
3
answers
558
views
$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns
I am interested in answering the following question:
Question
For a given $n$, does there exist a $2n \times 2n$ matrix with entries in $\{1, 0, -1\}$ having orthogonal rows and columns with exactly $...
- 539
7
votes
0
answers
224
views
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 ...
- 745
3
votes
0
answers
185
views
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 ...
- 11.2k
1
vote
1
answer
99
views
Orthogonal invariance of (weighted) Laplacian
It is well-known that if $O$ is an orthogonal map, then $\Delta u(Ox) = \Delta u$ where $\Delta$ is the Laplacian.
Now, let $A$ be a constant invertible matrix, then we define the weighted Laplacian
$$...
- 124
3
votes
1
answer
255
views
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 ...
- 161
1
vote
0
answers
188
views
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 ...
- 201
2
votes
1
answer
166
views
Subgroups of $\mathrm{SO}(A_0, \mathbb{F}_p)$
Let $n \geq 3$. Let $A_0$ denote the $n \times n$ symmetric matrix with $1$'s on the antidiagonal and $0$'s everywhere else. We can define the associated special orthogonal group
$$ \mathrm{SO}(A_0, \...
- 1,570
2
votes
0
answers
194
views
Maximize the product of Hadamard matrix and a vector
Let $X$ be an $n \times n$ Hadamard matrix (i.e. entries are in $\{-1,1\}$ and rows are orthogonal). For my application, we can assume $n=2^k$.
Given a vector $\bf{w} \in R^n$, I want to find the $X^*$...
- 21
7
votes
2
answers
373
views
Proving a lemma for a decomposition of orthogonal matrices
Setting
Consider two independent orthogonal matrices, which are decomposed into 4 blocks:
\begin{align}
Q_{1}
=
\left[\begin{array}{cc}
A_{1} & B_{1}\\
C_{1} & D_{1}
\end{array}\right]
,
\,Q_{...
- 673
3
votes
0
answers
156
views
Matrix equation and spherical harmonics
I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...
- 69
3
votes
1
answer
320
views
The invertible matrices $S$ that satisfy $A=SDS^T$
Any real symmetric matrix $A$ can be written as $A=SDS^T$ for some diagonal matrix $D$ and invertible matrix $S$. Let's fix $D$ to be the (diagonal) inertia matrix of $A$, which has an entry $1, -1, 0$...
- 980
3
votes
1
answer
275
views
Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on $O(n)$?
Denote by $d_C(\sigma,\mu)$ the minimal number of transpositions needed to go from a permutation $\sigma$ to a permutation $\mu$. E.g. if $d_C(\sigma,\mu)=0$, then $\sigma=\mu$, if $d_C(\sigma,\mu)=1$,...
- 131
2
votes
1
answer
841
views
Matrix derivative w.r.t. a general inverse form: $(A^TA)^{-1/2}D(A^TA)^{-1/2}$
I want to find derivative of matrix $(A^TA)^{-1/2}D(A^TA)^{-1/2}$ w.r.t. $A_{ij}$ where D is a diagonal matrix. Alternatively, it is okay too to have
$$\frac{\partial}{\partial A_{ij}} a^T(A^TA)^{-1/2}...
- 519
1
vote
0
answers
72
views
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 $f:V^2 \to \mathbb R$, $(...
- 6,853
12
votes
3
answers
383
views
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 ...
- 223
2
votes
1
answer
164
views
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{\...
- 6,741
1
vote
0
answers
142
views
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 : $ADB^{T}$, ...
- 11