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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Inequality for a positive matrix

I consider an $n\times n$ real positive-definite matrix $A$, which I rewrite in its diagonalized form: $A=O^TDO$, where $O$ is an orthogonal matrix and $D$ is diagonal. Since $A$ is positive-definite, ...
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Generate a uniformly sampled orthonormal matrix that 'rotates' $k$ vectors $x_0 \in \mathcal{R}^{n \times k}$ into $y_0 \in \mathcal{R}^{n \times k}$

We know that orthonormal matrices $H \in \mathcal{R}^{n \times n} $ are rotation matrices. Is there a general method to uniformly generate rotation matrices that can rotate a given set of vector $x_0 ...
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Antisymmetric Matrices and Orthogonality

My notes state: Given an orthogonal $n\times n$ matrix $A=I+pB$, where $I$ is the identity matrix, $p\ne 0$ is a real number and $B$ is an $n\times n$ matrix, then $B$ is skew-symmetric. I know that $$...
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Possibility of constructing an orthogonal matrix

I have a $n \times n$ matrix $A$, and for each column $j$ of $A$, $\sum_{i}(A[i,j])^2 = 1$ holds. (i.e. sums of squares of each column adds to $1$.) I want to build a new $2n \times 2n$ matrix that ...
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Decomposition of a 4D rotation into a particular sequence of simple rotations.

It is known that any rotation in $SO(n)$ can be decomposed into a particular sequence of $n(n-1)/2$ simple rotations (that is, rotations which rotate a 2D plane in $\mathbb{E}^n$). The procedure to ...
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Non compact group of quadratic matrices [closed]

Currently, I am working through Problems and Solutions for Groups, Lie Groups, Lie Algebras with Applications. Right in the first Lie Group Problem (p.73) it is asked to prove that the orthogonal ...
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Are the determinants of a matrix and the diagonal matrix obtained after diagonalization equal? [closed]

This question is related to a derivation step needed to find an n-dimensional generalization of the Gaussian integral, derived here: reference for multidimensional gaussian integral Is it true that ...
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If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2).$

Prove or provide a counterexample: If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2),$ where $\mathrm{tr}$ denoted the trace and $\mathrm{rk}$ the ...
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Any element of $SO(n)$ decomposes as a product of $n(n-1)/2$ elements, each an exponential of mutually linearly independent generators?

Physics person here, so this might be a simple question that has a straightforward answer in some subfield of math that I am not aware of. Thanks in advance! We are given an arbitrary set of $n(n-1)/2$...
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What kinds of processes do orthostochastic matrices represent?

An orthostochastic matrix is a bistochastic matrix $B$ such that $B_{ij}\equiv O_{ij}^2$ for some orthogonal matrix $O$. Bistochastic matrices can be given a direct interpretation as describing ...
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Resources for finding the dimension of the orbits

I am working on a problem concerning a Lie group acting on a smooth manifold or more generally, groups acting on topological manifolds or topological spaces. I am wanting to become more familiar with ...
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Question on the connectedness of the orthogonal group

I want to show that the quotient $O_2^- = O_2/SO_2$ is connected. My idea was as follows: It's easy to show that $SO_2$ is connected. $S0_2$ is a topological group (normal subgroup of a topological ...
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Can I get some help reading a proof for classification of finite subgroups of $SO(3)$? There's one line I can't understand.

The proof that I'm reading. Here, we have $$\left|G \right|-1=\frac{1}{2}\sum_{p \in P} \left ( \frac{\left| G\right|}{\left|\text{Orb}(p) \right|}-1 \right )=\frac{1}{2}\sum_{O}\left ( \left| G\right|...
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Change of eigenvalues under near orthogonal matrix multiplication

Let $A,B\in\mathbb{R}^{d\times d}$ be positive definite diagonal matrices. Let $\Phi\in\mathbb{R}^{n\times d}$ satisfy $\text{rank}(\Phi) = n \leq d$ and be such that $\Phi\Phi^T = I_n$. When $d = n$, ...
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Why the Euclidean Group $E(n)=\left\{Ax+b|A\in O(n),b \in T(n) \right\}$?

I'd like to show $$E(n)=\left\{ f |f(x)=Ax+b \ \text{where} \ A\in O(n),b \in T(n) \right\}$$  To be more precise, $E(n)$ is the isometry group of $\mathbb{R}^n$, which is also known as $I(\mathbb{R}^...
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Smooth traversal of 𝑆𝑂(𝑛)

I am trying to constrain the space of matrices used for the layers of a neural network to those in 𝑆𝑂(𝑛). It is proven that 𝑆𝑂(𝑛) is a manifold. I'm trying to find a way to smoothly traverse ...
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Which orthogonal rotation matrices for diagonalisation

We consider 2D orthogonal rotation matrices $R$. We consider a real matrix $$A = \begin{pmatrix} a & b\\ b & c \end{pmatrix}.$$ I write that $B = RAR^T$ for some diagonal matrix $B$. I would ...
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Show there exists a constant $C>0$ such that $\forall x\in\mathbb{R}^{n},x^{t}Ax\leq C\cdot\left\Vert x\right\Vert ^{2}$

Question Let $A\in M_{n\times n}\left(\mathbb{R}\right)$ be a square symmetric matrix. Show that there exists constant $C>0$ such that for any $x\in\mathbb{R}^{n},x^{t}Ax\leq C\cdot\left\Vert x\...
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Universal Properties of Orthogonal Matrices

I wanted to ask this question since I have seen conflicting viewpoints on it. Are orthogonal matrices necessarily symmetric? I do not believe so but some website said they were so I need to confirm. (...
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Similarity relationship between orthogonal maps of the diagonal representation of $A + B$

Question: What can be said in general of the relationships of diagonalization similarities between $A + B$ and its summands? For example consider $f_{AB}(t) := t A + (1-t) B $ a function that ...
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Eigenvectors of an Orthogonal Matrix

It is true that for orthogonal matrices, all eigenvectors of a single eigenvalue are orthogonal to all other vectors of different eigenvalues(distinct eigenbases are orthogonal). My question is are ...
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Decomposition of orthogonal complex matrix

Note : $\mathrm{O}(n)$ is $\mathrm{O}(n, \mathbb{R})$. I tried to solve the following exercise : For every complex orthogonal matrix $g \in \mathrm{O}(n, \mathbb{C})$, i.e. $g^{-1} = {}^\mathrm{t}g$, ...
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How to understand the relationship of the fundamental subspaces in these big pictures?

I am struggling like 6 hours to understand what this content in the middle mean. Can u get me some clue to interpret it? I understand all the stuff on the sides. So, row space is perp. to null space ...
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Spectral radius of a specific matrix multiplied by a diagonal matrix $D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$

Let $A=(a_{ij})\in M_n(\mathbb C)$, $n\geq 3$ be a matrix satisfying : $\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$ for all $i=1,\ldots n$. $\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$ for all $j=1,\ldots n$. ...
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A matrix is orthogonal if its similar matrix is orthogonal

I'm wondering how to prove this statement. Say $A=S^{-1}BS$ and $A^TA=I$, then $(S^{-1}BS)^T S^{-1}BS = I$; so $ S^TB^T{S^{-1}}^{T} S^{-1}BS = I$, but I'm not sure about the next step.
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How to prove: A geometric characterization of a $2 \times 2$ orthogonal matrix $A$ with $\mbox{det}(A) = 1$.

In texts on linear algebra, a standard example for a $2 \times 2$ orthogonal matrix is the rotation matrix given by $$ R = \left[ \begin{array}{cc} \cos \theta & - \sin \theta \\ \sin \theta &...
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Fast Multiplication of matrix and vector

Multiplication of the DFT matrix and any vector can be implemented by FFT. I'm interesting about other fast Multiplication. Suppose there are three matrix of size $N\times N$, $F$ , $Q$ and $P$, where ...
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SVD of an orthogonal projector

Here is my observation: Suppose there is an orthogonal projector $P$ such that $P=P^2$. Then for arbitrary $x$, $Px$ and $(I-P)x$ are orthogonal. So we have $$ x^* P^* (I-P)x=0$$ where $A^*$ means ...
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Gram Schmidt process against orthogonal basis W

Another question that has a wrong answer from people adopting it. Am I wrong or the textbook wrong? Answer from book: My ans using the Gram-Schmidt process: such that $\vec{x_1} and \vec{x_2}$ are ...
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Understanding orthogonal projection of $\vec{y}$ on to span of orthogonal set, with an example

This is to verify if there's an issue with my understanding or if there's issue with the textbook. There seem to be also a previous question here on exactly the same, hoping to help myself and future ...
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Differential form computation

I was reading a text on differential geometry and I noticed this: "Given $\omega=xdy\wedge dz+ydz\wedge dx+zdx\wedge dy\in\Omega^2(S^2)$ with $S^2=\{p\in\mathbb{R}^3:\lvert p\rvert=1\}$, show ...
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Matrix representation of a reflection

Assume I have a vector $v =$ $(-1, 1, -1)^T$ and $A$ is the relection through the plane orthogonal to $v$. How would I find the matrix representation of A? I have in my notes that the general ...
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Orthogonal Matrix with respect to different inner products

From what I understand, an orthogonal matrix is one that satisfies $A\cdot A^t = I_n$. In such case, from what I saw online, $\forall x\in \mathbb{R}^n,\;\left\Vert Ax \right\Vert =\left\Vert x \right\...
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Conditions for orthogonality of Jacobi matrices

Under which conditions is a Jacobi matrix of a coordinate transformation orthogonal? Background: I investigate one of the invariants of a rank two tensor under coordinate transformation with a ...
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The conjugation action $\mathbb{H}^*\times \mathbb{H} \rightarrow \mathbb{H}$ restricted to unit-quaternions yields an orthogonal representation

Consider the action $\mathbb{H}^*\times\mathbb{H} \rightarrow \mathbb{H}, (h,h')\mapsto hh'h^{-1}$. Show that it preserves the orthogonal-decomposition $\mathbb{R}\bigoplus $Im$\mathbb{H}$, and ...
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Question regarding the Three orbit case of classification of finite subgroups of $SO(3)$. (Tetrahedral Group)

When we have three orbit case, we would specifically have the subcase such that $r_1,r_2,r_3=2,3,3$ where each $r_i$ represents the order of a stabilizer of some pole $p_i$. If we let the finite ...
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Curve with a prescribed Frenet frame

Suppose you are given an antisymmetric matrix $X$. Then $A(t)=e^{Xt}$ is a curve of orthonormal matrices, with $A(0)=Id$. Is it possible to construct a curve whose Frenet frame vectors $T,N,B$ are the ...
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Questions regarding a proof of "classification of the finite subgroups of $SO(3)$".

I'm self-studying the classification of finite subgroups of $SO(3)$ with this paper. On page 13 of the paper, there is a paragraph (below the equation (9.3)) that I'm having trouble understanding. The ...
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Is there a name for this 'splitting' of an orthogonal structure into unitary and symplectic structures?

Suppose we have a (non-trivial) representation of some special orthogonal group $SO(p,q)$ over a real vector space $V$, I.e. the action of elements of $SO$ leave invariant a non-degenerate symmetric ...
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Commutativity of Multiplying Semi-orthogonal Matrix with Symmetric Matrix

I encounter the following question when I read the paper "Quadratic Optimization with Orthogonality Constraints" (https://arxiv.org/pdf/1510.01025.pdf). Simply speaking, let $X\in \mathbb{R}^...
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QR decomposition and a change of basis

Consider a square matrix $A$ and its $QR$ decomposition $A = QR$, where $Q$ is orthogonal and $R$ is upper triangular matrix. Now consider a change of basis. Let $A' = PAP^T$, where $P$ is an ...
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If a linear map sends orthonormal basis on orthonotmal basis then it is an isometry?

Let $(\mathbf{u_1,u_2,u_3})$ and $(\mathbf{v_1,v_2,v_3})$ orthonormal lists. Define $T:\mathbb{R^3}\rightarrow \mathbb{R^3}$ through the lineal extension $T(u_k)=v_k$. Is T an isometry? My attempt: I ...
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Quadratic forms and orthogonal basis

I have the quadratic form $q(x,y,z) = x^2 - 2y^2 +xz +yz$ Give the polar form and the matrix of q in canonic basis. I think this question is ok : $$ A = \begin{pmatrix} 1 & 0 & 1/2 \\ 0 &...
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When $A \in SO(3)$, $A$ is always a rotation.

I have official proof of this problem, but I am having trouble understanding some parts of it. Thus, I would like to share the parts and would like to get checked if my understanding is correct. ...
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Number of independent elements of an orthogonal matrix

I don't understand the following fact. The $n^2$ elements of an orthogonal matrix $A$ of order $n \times n$ are not independent. This follows from the fact that $A′A = I_{n}$ which implies that the ...
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How to show that there is a non-zero $v \in \mathbb{R}^n$ for any $A \in O(n)$ s.t $Av=\pm v$ whenever $n$ is odd.

Intuitively, this makes sense, but just in $\mathbb{R}^3$. For example, in $\mathbb{R}^3$, when any $A$ is given, this $A$ determines how much should a point in $\mathbb{R}^3$ be rotated about the ...
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Decomposition of a matrix with orthogonal columns

Note that this question is also posted in mathoverflow. I'm new to the community so I'm also posting this here if it is more appropriate. I am given a square matrix $A = [A_1, A_2, ..., A_n]$, where ...
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Covariance of entries of a random orthogonal matrix

Let $Q$ be an $n\times n$ matrix that follows the Haar measure (that is, we begin with an $n\times n$ matrix of standard normal random variables and perform Gram-Schmidt orthogonalization) I am trying ...
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Confusion about connection between orthogonal matrices and rotation in higher dimensions

I found this question which discusses that all orthogonal matrices are rotations/reflections, since the map $X\rightarrow AX$ preserves the scalar product, with $A$ an orthogonal matrix (see proof in ...
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Meaning of maximisation of the trace of a product of matrices

Suppose I $$\begin{array}{ll} \underset{\Omega \in \Bbb R^{n \times n}}{\text{maximize}} & \text{Tr}(\Omega A)\\ \text{subject to} & \Omega^\top \Omega = I_n\end{array}$$ What am I looking at ...
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