Questions tagged [orthogonal-matrices]
An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose.
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Need explicit formula for certain "$q$-numbers" involving gcd's
The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
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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 ...
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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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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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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
$$\...
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Differentiable functions on $\mathbb{R}^n$ whose derivative is everywhere a scalar multiple of a special orthogonal matrix
The Cauchy–Riemann equations say that if $u : \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic then, regarded as a linear transformation of $\mathbb{R}^2$, its derivative is either zero or, up to a ...
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Matrix equation and spherical harmonics
I have a set of functions expanded in a finite number of spherical harmonics (up to degree $L$),
$$
\eta_k^n(\theta,\phi) = \sum_{l=0}^L \sum_{m=-l}^l d_{kl}^{nm} Y_l^m(\theta,\phi)
$$
Similar to the ...
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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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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)...
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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, ...
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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 ...
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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^*$...
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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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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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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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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 ...
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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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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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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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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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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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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}^{...
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How can one orthogonalize the pointwise sum of two orthogonal sets?
Let $n = 2k$, and suppose that $V = \{v_1, \cdots, v_k\}$ is an orthogonal set in $\mathbb{R}^n$. In other words, the vectors in set $V$ are pairwise orthogonal to each other.
Now, consider a new set $...
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Smallest Singular Value of submatrices of a column-orthogonal matrix
Suppose we have a column-orthogonal matrix $\mathbf {U}\in\mathbb{R}^{n\times p}$, satisfying $\mathbf {U}^{\top}\mathbf {U}=\mathbf {I}_p$. We select $m<n$ rows of $\mathbf {U}$ randomly and get $\...
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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 $...
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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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Applying a linear operator to a basis set following SVD orthonormalization
Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases.
My intention is to:
decompose $\Phi$ via SVD:
$U \Lambda V^T = \Phi$
to create it's ...
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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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