Questions tagged [orthogonal-matrices]

Matrices with orthonormalized rows and columns. An orthogonal matrix is an invertible real matrix whose inverse is equal to its transpose. For complex matrices the analogous term is *unitary*, meaning the inverse is equal to its conjugate transpose.

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How to prove Vol_O(n)=2^nVol_O_+(n)?

Let $O(n)$ be the orthogonal group and $DO(n)$ be its discrete subgroup of diagonal matrices with $\pm 1$s on the diagonal. The metric on $O(n)$ is induced by Euclidian metric on $M(n)$. Let $O_+(n)=O(...
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Gram schmidt swapping two vectors

The question has background here but it's really just a linear algebra question. Suppose I have $B = (b_1,\cdots,b_n)$ vectors and I perform Gram Schmidt process (with no normalization of vector) ...
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a group isomorphic to $O(2N)$

Consider the set of $2N\times 2N $ complex matrices $T$ satisfying the conditions $$TT^\dagger = I_{2N}$$ and ($^*$ means taking the complex conjugate) $$ T^* = \gamma T \gamma ,$$ where $\gamma $ is ...
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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 ...
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A supremum on orthogonal matrices

I'm working on a problem where I want to find the supremum over the orthogonal group $O_n(\mathbb{R})$ of the sum of the upper triangular elements of matrices in this group, specifically we want to ...
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How would one integrate over $SO(n)$?

Suppose you want to find the average of an $n\times n$ diagonal matrix $A$ over all possible rotations, $$ \langle A\rangle = \int\limits_\text{SO($n$)} Q^T A Q \; dQ. $$ It's easy enough to do this ...
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Conditional distribution of random orthogonal projection matrix

I have encountered a rather curious question. Suppose I have a symmetric idempotent orthogonal projection matrix $A\in\mathbb R^{N\times N}$ that projects onto a uniformly random $n-$dimensional ...
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Orthogonal transformation of Heteroskedastic matrices

Consider two $N \times N$ dimensional real matrices $A$ and $B$. $A$ is a diagonal matrix with all non-zero elements taken from a real Gaussian distribution with mean $\mu = 0$ and variance $\sigma = \...
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Distribution theory for random projections

Suppose $v$ is a fixed vector in $\mathbb R^n$, and let $u\in S^{n-1}$ (unit sphere in $n$ dimensions; $S^{n-1}=\{x\in\mathbb R^n:\|x\|=1\}$) be uniformly generated. What is the distribution of $\...
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Centralizer generators

This is a question posted on overflow but no reply has been received. In the algebraic group $G=\operatorname {PSO}(4,K)<\operatorname {PCGO}(4,K)$ where $K$ is an algebraically closed field of an ...
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Can reflections always be represented as rotations in higher dimensions?

If we think of reflection in $\mathbb{R}^1$ (multiplication by $-1$), this can be represented as $180$ degree rotation in euclidean plane (assuming a "natural embedding" notion of $\mathbb{R}...
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Construction of orthogonal matrices from orthogonal polynomials

Let $\{P_n\}_{0\le n\le N}$ be the family of orthogonal polynomials associated with the following inner product: $$ \langle f, g \rangle_{1} = \sum_{k=0}^{N}{f(k)g(k)w_1(k)} $$ where $ w_1(\cdot)$ is ...
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Skew-symmetric matrices and the fact that their exponential matrix is orthogonal

I want to ask why "If $A$ is skew-symmetric ($A^T=-A$) then $e^{At}$ is an orthogonal matrix". Here is my solution step: $e^{At}*(e^{At})^T =e^{At}*e^{-At}=e^{At-At}=e^0$ I think the matrix $...
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Relations between row and columns of orthogonal matrix

Given a orthogonal matrix $Q \in \mathbb{R}^{n\times n}$, we know $Q^{\top}$ is also orthogonal. Let $Q$ represent a linear transformation from an Euclidean space to itself, then reading from the ...
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Given $\|x_1\|=\|x_2\|$, then there is an orthogonal matrix $\Gamma$ such that $x_2=\Gamma x_1$.

Let $x_1, x_2$ be members of $\mathbb R^p$ such that $\|x_1\|=\|x_2\|$. Then there is an orthogonal matrix $\Gamma$ such that $x_2=\Gamma x_1$. How to prove the above? I know given $x_2=\Gamma x_1$, ...
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What do the parts of the Gram-Schmidt process mean and represent in space?

I am struggling to understand what the different parts of the Gram-Schmidt process represent. Suppose we have a basis $\{x_1, x_2\}$ We would then find a orthogonal basis by doing the following : $$...
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Striving towards a more precise, truthful, and accurate understanding of why gimbal lock happens. [closed]

Please, somebody explain why gimbal lock happens properly. Not a single correct solution out there, people always seem to gloss over the important details and if you dig into the common explanations, ...
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A probelm of constructing an orthogonal matrix and $n$ positive real numbers

$A \in M_n (\mathbb{R})$ and $A$ is invertible, let $S=\left\{(x_1,x_2,\cdots,x_n)^{'} \in R^{n}\mid\sum_{k=1}^{n}x_{k}^{2}=1\right\}$ and $T=\left\{Ax \mid x = (x_1,x_2,\cdots,x_n)^{'} \in S\right\}$,...
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Is every matrix that sends spheres to spheres of the same radius an orthogonal matrix?

Matrices send spheres to ellipsoids. Orthogonal matrices send spheres to spheres of the same radius. Is the converse true? Is every matrix that sends spheres to spheres of the same radius an ...
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Finding the Vector $v$ for a Given Householder Matrix Transformation of Non-Collinear Vectors $a$ and $b$

Consider a vector $v$ in $\mathbb{R}^{n\times1}$. The Householder matrix is defined as follows: $$H(v)=I-\dfrac{2vv^T}{v^Tv}.$$ It can be demonstrated that $H(v)$ is symmetric and orthonormal. The ...
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Set of Rotation Matrices in $R^n$ Which Fix (1,1,…,1) [closed]

For the case where $n=3$, I was given these rotation matrices which fix the vector $(1,1,1)$. How would I generalize these matrices for $R^n$?
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Number of rational points of orthogonal groups

Let $p$ be a prime and $q$ be a power of $p$. Let $k$ be an algebraic closure of $\mathbb F_q$, the field of $q$ elements. Let $N$ be an integer $\geq 2$, and $G = GL_N(k)$. There is an involutive ...
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A question related to the action of a group on the real projective space

I am trying to make sense of the following proof showing that $\mathbb{P}\simeq \text{O}(n+1)/(\text{O}(1)\times \text{O}(n)),$ where $\mathbb{P}:=\mathbb{P}^n(\mathbb{R})$ is the $n$-dimensional real ...
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Does this condition imply orthogonality?

In transforming matrices between bases I have come across this curious equation: $\mathbf{A} \mathbf{B} \mathbf{C} = \mathbf{A}^{\mathsf{T}} \mathbf{B} \mathbf{C}^{\mathsf{T}}$, for all $\mathbf{B}$. ...
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Existence of Orthogonal Symplectic matrix

I am investigating the existence of an orthogonal matrix satisfying specific conditions. Let $\mathbf{v}_1, \dots, \mathbf{v}_M \in \mathbb{R}^{2n}$ be real vectors with unit norm, where $M \leq 2n$. ...
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How unique is the orthogonal diagonalization of a real symmetric matrix, if we don't change the diagonal matrix of eigenvalues (no permutaiton)?

Let $A$ be a real symmetric $n\times n$ matrix, so it's orthogonally diagonalizable, i.e. there is $P\in O(n)$ so that $P^{-1}AP=P^{T}AP=D$(diagonal). I'm asking myself: how unique can $P$ be? Here ...
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Does Matrix projection impacts centralization

I have a dataset that each sample is a matrix $A_{m\times n}$, and I have N samples in my dataset. I want to centralize my data across my samples, meaning that for each element in the matrix, I want ...
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self-orthogonal binary codes

From Robert Griess's article "Elementary abelian p-subgroups of algebraic groups": (2.7) Definition. Let char$(\mathbb{K})\neq 2$ and let $V$ be an $m$-dimensional vector space with ...
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Showing that two orbit spaces are isomorphic

Consider the following two group actions: $O_n\times Sym_n(\mathbb{R})\rightarrow Sym_n(\mathbb{R})$ Given by $(A,B)\mapsto ABA^{-1}$, where $O_n$ denotes the group of orthogonal matrices and $Sym_n(\...
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Multiply orthogonal matrix not change eigenvectors?

Multiplying an orthogonal matrix not change eigenvectors (directions)? I think it is true in some cases. The orthogonal matrix represents rotations and reflections. Eigenvectors do not change ...
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Lie algebra of real unitary matrices

The Lie algebra associated to the group $SO(n)$ of real-valued special orthogonal matrices, is given by the set $\mathfrak{so}(n)$ of anti-symmetric real-valued matrices equipped with the commutator. ...
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What matrices $M$ satisfy $M_{il} M_{jm} M_{kn} \delta_{lmn} = \delta_{ijk}$?

I'm trying to better understand matrices that are defined by preserving some other tensor. I recently asked a more involved question, and I realized it might be better to learn simpler examples first. ...
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Decompose a orthogonal map as orthogonal maps in two directions

Let $R \in O(N)$ be an orthogonal map in $\mathbb{R}^N$. Write $N = N_1 + N_2$. I am wondering if is it possible to find a map $T \in O(N_1)$ and $L \in O(N_2)$ such that $R(x,y) = (T(x), L(y))$. If ...
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What is known about merely-orthogonal matrices?

I'm interested in square matrices whose columns are orthogonal, but not necessarily orthonormal, non-zero vectors. Answers to other questions on this topic have noted that such matrices do not have an ...
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Why projection onto the Stiefel manifold fails to solve the orthogonal Procrustes problem

The orthogonal Procrustes problem finds an orthogonal matrix $\Omega$ minimizing the Procrustes objective: $$ \min_\Omega ||\Omega A - B||_F, \quad \Omega^\top \Omega = I $$ It is well known that the ...
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A $3 \times 3$ matrix with unit row and unit column but all unique elements must be orthogonal matrix.

I am using basic matrix theory to prove that if a $3 \times 3$ matrix has unit rows and unit columns but all entries are unique then matrix must be orthogonal. But I have trouble proving this. All I ...
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Is it possible to find irreducible representations of $SO(3)$ without passing to lie algebras?

Forgive my lack of rigor because I am professionally not into science/math. Is it possible to find irreducible linear(in contrast to projective) representations of $SO(3)$ without passing to lie ...
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How to determine if the orthogonal projection is onto a line or a plane?

I'm trying to determine if the projection using the $P$ matrix is onto a line or a plane, if the matrix is given as: $$P = \frac{1}{3} \cdot \pmatrix{2 & 1 & 1 \\ 1 & 2 & -1 \\ 1 & ...
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Procrustes Problem With Scaling

Given matrices $A,B \in \mathbb{R}^{m,n}$, the orthogonal Procrustes problem asks to find an orthogonal matrix $Q \in \mathbb{R}^{n \times n}$ such that $$\|AQ - B\|_F^2$$ is minimized. There is a ...
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maximal subgroup of the general linear group

Maybe I'm being silly. As algebraic groups over an algebraically closed field $K$ of characteristic not $2$, is the conformal orthogonal group $CO_{2n}$ a maximal subgroup of $GL_{2n}$(or $CO_{2n+1}$ ...
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Why is $SO(n)$ a smooth manifold [duplicate]

I am new to differential topology and a question that I came across was to show that $SO(n)$ is a smooth manifold of dimension $\frac{n(n-1)}{2}$. The dimension follows from the fact that the columns ...
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How to choose a vector which is linearly independent from a set of orthogonal vectors?

I have a non-complete set of orthogonal vectors $V=[\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n]$ with $m > n$ entries. I would like to choose another vector $\mathbf{w}$ which is linearly ...
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Standard Matrix for an Orthogonal Projection, Explain why $AA = A$. [duplicate]

I have this question on a homework problem. Let $A$ denote the standard matrix for an orthogonal projection of the plane $\Bbb R^2$ onto a line $L$ through the origin. Explain why $AA = A$. Can ...
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For orthogonal matrix $P = [P_1 \vert P_2]$, show that $col(P_1)^{\bot} = col(P_2)$

I'm currently struggling to solve this question. The first one $col(P_2) \subset col(P_1)^{\bot}$ is quite straightforward ($P_1$ is ($n \times r$) and $P_2$ is ($n \times (n-r)$)) $x \in col(P_2) \...
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General linear group inclusion

Do we have $\operatorname{GL}(n,F)\le \operatorname{O}(2n,F)$ where O means general orthogonal group and $F$ is an algebraically closed field? I checked some finite group cases: $\operatorname{GL}(2,5)...
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Special orthogonal matrices, Geodesic and Manifold

My task is to find some manifold on stoichiometric matrices and find the geodesic distance between these nodes (not the euclidean distance). Here's my idea so far: Suppose we are given a matrix N of ...
meatball2000's user avatar
2 votes
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Finite subgroups of O(3)

My question is the following: What are the finite subgroups of O(3), the group of linear isometries? I managed to find a lot of good references describing the finite subgoups of SO(3) (only direct ...
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Concatenated diagonalization of combined real/imaginary diagonalization

I have a problem involving a complex matrix that I need to diagonalize and apply weights to the entries according to a function relating only to the eigenvalues of the real part of the matrix. Suppose ...
George Kyriakou's user avatar
1 vote
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Which affine transformations preserve rectangles?

I need a way to decide if a given affine transformation preserves arbitrary rectangles in $\mathbb{R}^2$, meaning after applying it to any rectangle, it is still a rectangle afterwards. Thought ...
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Rotation matrix and orthogonal matrix

If I am given an orthogonal matrix $A$ and I right multiply by $W $ to get $B=AW$ where $W$ is a diagonal matrix then essentially, I am scaling the columns of $A$ by the diagonals of $W$, preserving ...
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