Questions tagged [orthogonal-matrices]
An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.
125 questions
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Efficient construction of an orthogonal matrix with prescribed diagonal quadratic forms
Let $\boldsymbol{\lambda}=(\lambda_1,\dots,\lambda_n)$ and $\mathbf{d}=(d_1,\dots,d_n)$ be two vectors of real numbers sorted in non‑increasing order, satisfying the majorization conditions
$$
\sum_{i=...
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Explicit orthogonal matrix with prescribed diagonal for the path graph Laplacian
Let $N \ge 2$ and consider the diagonal matrices
$$
\mathbf{D} = \operatorname{diag}(\lambda_0,\lambda_1,\dots,\lambda_{N-1}), \qquad
\mathbf{D}_0 = \operatorname{diag}\!\left(0,\frac{4}{N},\frac{8}{...
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For which $n$ is the minimum max entry of an $n$ by $n$ orthogonal matrix known? Is $n=3$ already an open problem?
Crossposted on Mathematics SE, where the question Orthogonal matrices with small entries was brought to my attention, though it is about bounds rather than exact values.
Let $\| A \|_{\max} := \max\...
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Hadamard determinant problem and the special orthogonal group
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PL{PL}$I was trying to reduce the Hadamard problem of calculating the maximum value of the determinant of a $\{1,-1\}$-matrix to the problem of ...
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System of equations with an orthogonal matrix
I have six unit vectors $a_k$ and $b_k$ for $k \in \{1,2,3\}$. These are randomly drawn and are of dimension $3\times 1$.
Let $Q$ be an orthogonal matrix. I have noted that there are always$^\color{...
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Gram-Schmidt orthogonalization of $[i^{j-1}]$, and non-trivial zero elements inside
Consider the sequence of matrices $[i^{j-1}]_{(i,j)\in n\times n}$ Gram-Schmidt orthogonalized, the first ones are as follows, up to a normalization constant for each column. apologize if that's too ...
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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
$$
\underset{P^2 = P,\; \text{rank}(P) = p}{...
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Matching matrix columns under scaling, translation and orthogonal transformation
Given vectors ${\bf p}_1, {\bf p}_2, \dots, {\bf p}_n \in {\Bbb R}^d$ and ${\bf q}_1, {\bf q}_2, \dots, {\bf q}_n \in {\Bbb R}^d$, define the $d \times n$ matrices
$$ {\bf P} := \begin{bmatrix}
...
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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, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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