Questions tagged [orthogonal-matrices]
An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.
118 questions
2
votes
1
answer
197
views
Do top eigenvectors maximise both Tr$(P\Sigma)$ and Tr$(P\Sigma P\Sigma)$ for orthogonal projection matrices P?
Let $P \in \mathbb{R}^{d \times d}$ be an rank $p < d$ orthogonal projection matrix given by $P = VV^T$, $V \in \mathbb{R}^{d \times p}$. Do we have that
$$
\underset{P^2 = P,\; \text{rank}(P) = p}{...
- 23
1
vote
0
answers
167
views
Matching matrix columns under scaling, translation and orthogonal transformation
Given vectors ${\bf p}_1, {\bf p}_2, \dots, {\bf p}_n \in {\Bbb R}^d$ and ${\bf q}_1, {\bf q}_2, \dots, {\bf q}_n \in {\Bbb R}^d$, define the $d \times n$ matrices
$$ {\bf P} := \begin{bmatrix}
...
- 11
2
votes
3
answers
258
views
Quantifier-free description of the orthostochastic matrices
What is a quantifier-free formula defining the space of orthostochastic matrices? More precise version below.
Fix $n \in {\Bbb N}$, let ${\mathcal M}_n$ denote the space of real $n\times n$ matrices, ...
- 688
10
votes
1
answer
635
views
Computing the exponential form of a unitary operator
Consider the unitary operator $U$ on the Hilbert space $H:=L^2([0,\frac\pi2])$, that takes $\cos((2k+1)x)$ to $\sin((2k+1)x)$, for $k\in\mathbb N$ (both are orthogonal basis). How can we explicitly ...
- 63.5k
0
votes
0
answers
130
views
Minimum eigenvalue and semidefinite cone
I have an linear matrix inequality(LMI) in the form: $G + x_1F_1 + \cdots + x_nF_n \succeq 0$, where $G$ and $F_i$ are symmetric matrices, $x_i \in \{0, 1\}$, and a matrix $A \succeq 0$ means that the ...
- 21
5
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 ...
- 173
3
votes
1
answer
300
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
268
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
244
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 ...
- 2,599
2
votes
1
answer
141
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
75
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, ...
- 542
7
votes
2
answers
271
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 ...
- 513
1
vote
1
answer
153
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
86
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 ...
- 71
2
votes
0
answers
125
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 ...
- 41