All Questions
Tagged with orthogonal-matrices matrix-theory
14 questions
3
votes
1
answer
191
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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 ...
- 256
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 ...
- 10.4k
1
vote
1
answer
455
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About the Hadamard conjecture
On the wikipedia article about Hadamard Matrix it says that "The smallest order that cannot be constructed by a combination of Sylvester's and Paley's methods is $92$"
But it also says that ...
- 696
8
votes
2
answers
519
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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 $\...
- 156k
2
votes
0
answers
104
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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
7
votes
0
answers
224
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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
6
votes
1
answer
986
views
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 ...
- 188k
5
votes
1
answer
826
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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+...
- 44.4k
16
votes
7
answers
2k
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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 ...
- 52.4k
2
votes
0
answers
79
views
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 $O$,...
- 2,712
3
votes
1
answer
416
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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 ...
- 28.6k
0
votes
1
answer
460
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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.
- 377
1
vote
0
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168
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Ring-theoretic version of a matrix problem
Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is ...
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