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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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264 views

How can I find the general form of an orthogonal matrix?

I know that the general form of orthogonal matrices is $$\begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$$ since they are all rotation matrices but how do ...
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131 views

Quaternion Converted to Rotation Matrix then Derived with Respect to this Quaternion

I was wondering how a derivative of a rotation matrix generated based on a quaternion and then differentiated with respect to this quaternion would be calculated. $$q_{1\times 4} = [q_0 q_1 q_2 q_3]^...
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31 views

Does the following unitary matrix factorization have a name?

I know any unitary matrix can be factored as follows: $$\underline {\overline {\bf{U}} } = \left( {\prod\limits_{j = N}^1 {\underline {\overline {\bf{\Psi }} } \left( {{{\underline {\bf{w}} }_j}} \...
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60 views

If the matrix representation of an orthogonal transformation with respect to a basis is an orthogonal matrix, the basis is an orthonormal basis?

I'm learning linear algebra and interested in the relationship between linear transformation, matrix representation and the basis. The followings are my questions: If the matrix representation of an ...
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105 views

Changing bases for orthogonal matrix

Hi I would like some help with the following exercise: We have an orthonormal basis $B_1$ and the vectors $$u_1= \begin{bmatrix}\frac{1}{3} \\ \frac{2}{3} \\ \frac{2}{3} \end{bmatrix} u_2= \begin{...
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72 views

Orthogonal Procrustes Problem Proof

I'm going through the proof for the Orthogonal Procrustes problem and I was able to derive all but the last few steps. Namely, the implication from lines $6 \Rightarrow7 \Rightarrow 8$ (I was able to ...
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25 views

Instability of even windings of SO(4)

Representatives of $\pi_1(SO(2))=\mathbb{Z}$ may be given by paths$$\theta\mapsto\left(\begin{array}{lr}\cos(n\theta)&-\sin(n\theta)\\\sin(n\theta)&\cos(n\theta)\end{array}\right).$$However, $\...
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146 views

Matrix norm/Frobenius norm inner product simplification

I'm going through the proof for the Orthogonal Procrustes problem and I wanted to see how they got the following relation. $$\langle \Omega A - B, \Omega A - B \rangle = ||A||^2_F+||B||^2_F-2\langle\...
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47 views

Projection onto the Set of Orthogonal Matrices - $ \mathcal{O}^{n} = \left\{ X \in \mathbb{R}^{n \times n} \mid {X}^{T} X = I \right\} $

Let $ \mathcal{O}^{n} $ be the set defined by $ \mathcal{O}^{n} = \left\{ X \in \mathbb{R}^{n \times n} \mid {X}^{T} X = I \right\} $, namely the set of Orthognal Matrices of size $ n \times n $. I ...
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100 views

Matrix inversion problem in ridge regression

The ridge estimator can be written in the following way, where the singular value decomposition of X is $X=UDV^{'}$. I can't quite figure out how the last step (4th step) was obtained from the 3rd ...
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201 views

SVD of orthogonal matrix and multiplying the left and right singular matrices?

I came across some code that takes some orthogonal matrix $R$ and computes the SVD of it $U, S, V^T = SVD(R)$ and then they compute the quantity $UV$ with the comment: ...
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39 views

$L_{1,1}$-norm and orthonormal matrices

Given any point $p \in \mathbb{R}^n$ and two matrices whose columns are orthonormal $X \in \mathbb{R}^{n \times j}$, $Y \in \mathbb{R}^{n \times (n - j)}$, such that $Y$ spans the orthogonal ...
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82 views

Formulas for Euler Axis components

I am trying to derive the formulas given in this book titled Atmoshperic and Spaceflight Dynamics by Ashish Tewari. I found questions related to finding the rotation matrices online but nothing that ...
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Whether $SO_k(\mathbb{Q})\subsetneq SO_k(\mathbb{Q(\sqrt{n})})$

Is it true that for all square-free $n$, and for all $k>1$, we have $SO_k(\mathbb{Q})\subsetneq SO_k(\mathbb{Q(\sqrt{n})})$? So far I have only discovered this: if $n$ has no prime factors $\...
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105 views

Parametrization of $SO(N)$ using unit vectors

The group $SO(N)$ has $m=N(N-1)/2$ generators and one thus needs $m$ angles to parametrize it. In the case of $SO(2)$ and $SO(3)$, one can also parametrize using unit vectors $\vec q$ in $\mathbb R^{...
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Prove that $HAH^{-1} = \left( \begin{smallmatrix} \lambda_{1} & b^{t} \\ 0 & B \\ \end{smallmatrix} \right) $

Consider $A \in \Bbb{R}^{nxn}$ with distinct eigenvalues $\lambda_{1}, ..., \lambda_{n}$ and respective eigenvectors $v_{1}, ..., v_{n}$. $H \in \Bbb{R}^{nxn}$ an orthogonal matrix so that $Hv_{1} = \...
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What does $A^{-1}=A^T$ have to do with “orthogonality”?

Whenever I read some use of the term “orthogonal”, I have been able to find some way in which it is at least metaphorically similar to the idea of two orthogonal lines in euclidean space. E.g. ...
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1answer
35 views

Find cross product of other part of orthogonal linear transformation of matrix

Say that I know a dense matrix $Y\in\mathbb{R}^{n\times r}$. Further, let $Q\in\mathbb{R}^{n \times n}$ be an orthogonal matrix and $Q^\top Y = (C^\top, D^\top)^\top$ be two dense matrices such that $...
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39 views

Eigenvectors of a vectorized tranpose matrix

Given the permutation matrix $P\in\mathbb{R}^{n^2}$ such that $P\mathop{\mathrm{vec}}(X)=\mathop{\mathrm{vec}}(X^T)$ for all $X\in \mathbb{R}^{n\times n}$, is there any closed form expression for the ...
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157 views

If an orthogonal matrix has determinant -1 then it has -1 as an eigenvalue

I want to show that if a real orthogonal matrix $A$ has determinant $-1$ then $\lambda=-1$ must be an eigenvalue of $A$. I have proven this in a long-winded way and I was wondering if these is a ...
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Number of equations in orthogonal matrix?

I have an orthogonal matrix $A^{n\times n}$, that is $A^TA=I$. What is the number of equations $A$ has to satisfy? It seems to me that the answer is $\frac{n}{2}(n+1)$. My reasoning as follows. Each ...
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81 views

Show that $\mathbf{M}^T\mathbf{M}^{-1}\left(\mathbf{M}^{-1}\right)^T\mathbf{M}=I$ where $\mathbf{M}= I - \mathbf{A}$ and $\mathbf{A}^T = - \mathbf{A}$

Let $\textbf{M} = I-\mathbf{A}$ (with $\mathbf{A}^T = -\mathbf{A}$) be a matrix and $\textbf{M}^{-1}$ its inverse. From the identities: \begin{equation} \textbf{M}^{-1}\textbf{M} = I\\ \...
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46 views

Find orthogonal complement and its basis

Let $U=\mbox{span}\left\{ \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}\right\}$...
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273 views

Prove that if $\Vert{Qx}\Vert = \Vert{x}\Vert$ then $Q^{-1} = Q^{t}$

The case where you have to prove $(Q^{t}Q)_{ii} = 1 \ \forall \ 1 \le i \le n$ is simple (you can choose $e_{i}$ as your $x$) but I am not able to show that $(Q^{t}Q)_{ij} = 0 \ \forall \ 1 \le i, j \...
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356 views

How to generate orthogonal matrices?

An orthogonal matrix has $\frac{n(n-1)}{2}$ degrees of freedom. Does there exist a function that maps $\frac{n(n-1)}{2}$ variables to the $n^2$ parameter orthogonal matrix? And can such a function ...
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30 views

Searching for orthogonal matrices with some property

I am searching for orthogonal matrices $A$ with the property $$|A|_F^2 = \deg( \chi_A(t) ) = 2 \deg( m_A(t)), tr(A) = 1$$ where $\chi_A(t)$ is the characteristic polynomial and $m_A(t)$ is the ...
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1answer
66 views

Orthogonal group representation induces an isometry on $L^p$ spaces

I am reading a paper and I am unsure about the following. I will first explain the setup. Suppose we have a measure space $(X,B,\mu)$. Let $G$ be a compact Lie group with Haar measure $\nu$. Suppose ...
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2answers
46 views

Show that $|A-I|=0$

let A be a $ n \times n$ orthogonal matrix,where n is an even number with $|A|=-1\quad$. Show that $|A-I|=0$ So basically I have to show that 1 is an eigenvalue of A. Here is how I proceeded:- ...
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208 views

Is the Euclidean distance from $\text{SO}(n)$ approximately convex?

Let $M_n$ be the vector space of real $n \times n$ matrices. Define $f:M_n \to \mathbb{R}$ by $f(A)=\text{dist}^2(A,\text{SO}(n))$, where the distance is measured using the Frobenius (Euclidean) norm....
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1answer
29 views

Sum of two rotations of the same matrix

Let $A$ be a symmetric matrix with real coefficients and let $O_1$ and $O_2$ be two real orthogonal matrices (i.e. $O_x^T O_x = I$). What can one say about $$ O_1^T A O_1 + O_2^T A O_2 $$ ? Is it ...
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1answer
101 views

Step in PCA proof - invariance of trace operator and orthogonality

I am stuck on the following step which is part of the derivation of Principal Component Analysis in the book "Foundations of Machine Learning" by Mohri, Rostamizadeh, page 283 Here is the context ...
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201 views

Direct sum of two subspace

Given the definition on textbook: Let $V$ be a subspace of $\mathbb R^n.$ Every vector $u \in \mathbb R^n$ can be written uniquely as $u = n + p.$ I still don't understand what it means because i am ...
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445 views

How to prove the complement $P^\perp$ of a projection matrix $P$ have relation $I-P=P^\perp$

I want to know how to prove that for a projection matrix $P$ and its complement matrix $P^\perp$. We have $$I-P=P^\perp$$ I do know the intuition that $P$ and $P^\perp$ project a vector into two ...
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1answer
96 views

Prove that if $M$ is orthogonal, then $\det(M)= \pm 1$

Recall that a matrix $M$ is orthogonal if it is square and $M^TM =I$. Prove that $\det(M) = \pm 1$ for every orthogonal matrix $M$. Not sure how to go about showing this for every orthogonal matrix
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Matrix Representation of Rotation in $\mathbb{R^3}$

I am reading the article: "The Banach-Tarski Paradox" by Karl Stromberg. At page no 153, author give a $3 \times 3$ matrix $\phi$ as follows: $$ \phi= \begin{pmatrix} -\cos\theta & 0 &...
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1answer
107 views

How many matrices does SO(2) contain?

Would I be correct in saying that the special orthogonal group SO(2) contains one matrix , namely; $A=\begin{pmatrix}{} \cos\theta& -\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}$ or ...
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0answers
21 views

Dot product over upper matrix and matrix of units

Does there exist a dot product on space of matrix nxn (n>1) regarding to which the matrix of all units would be orthogonal to any upper triangular matrix? First i thought to define dot product as $$(...
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1answer
191 views

Prove $O(2)$ is non-abelian?

$$O(2) = \{Q\in \mathbb{F}^{2\times 2} | Q^TQ= QQ^T=I\}$$ What is the most elegant way to prove that $O(2)$ is non-Abelian? Here is my thinking: I know that $O(2)$ can be generated by reflections and ...
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129 views

Rotationally Invariance Of Probability Density Functions

If $X_1,X_2$ are independent, normally distributed (with common variance) random variables, then the probability density function of the random vector $(X_1,X_2)$ in ${\bf R}^2$ is rotationally ...
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41 views

Help in proving linear algebra about inverse of a symmetric matrix

can anyone please help? Given: $\vec{x}$ - vector of rational numbers, $C$ - covariance matrix $C=(\vec x - \overline x)(\vec x - \overline x )^\top$ Prove: $C^{-1} =P^\top P$. Thanks
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Let $H$ be the set of matrices in $O_n(\mathbb{R})$ whose matrix entries are integers.

Prove that $H$ is equal to the set of all $n\times n$ matrices $A=(a_{ij})$ with integer coefficients such that $\vert a_{ij}\vert=0$ or $1$ for all $i,j$ and each row and column of $A$ has one non-...
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4answers
94 views

Find all possible values of $p$, $q$ and $r$ such that this matrix is orthogonal

Find all possible values of $p$, $q$ and $r$ such that the following matrix is orthogonal. $$B= \begin{pmatrix} \frac1{3}&\frac2{3}&\frac2{3} \\ \frac2{3}&\frac1{3}&-\frac2{3} \\ ...
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1answer
36 views

Show that the eigenvalues of $\mathcal{O}(n,\mathbb{R})$ have magnitude 1.

Suppose $A \in \mathcal{O}(n,\mathbb{R})$. Then $A^{T}A=AA^{T}=A^{*}A=AA^{*}=1$ (where * denotes the Hermitian conjugate). Thus $A$ is normal and hence by spectral theorem it has a decomposition such ...
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1answer
150 views

Derivative w.r.t orthogonal matrix

Let $A$ be an orthogonal matrix with elements $a_{ij}$ so that $\sum_k a_{ik} a_{jk} = \delta_{ij}$. I'd like to know what is $\frac{ \partial a_{ij} }{ \partial a_{kl}}$. If $A$ was a generic matrix ...
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1answer
54 views

projector on the boundary of a simplex. aka find basis for subspace $\sum_j x_j=0$.

I would like to find the (orthonormal) basis $\boldsymbol u=(u_1,\dots u_{n-1})$ of the vector subspace \begin{equation} \mathcal S^{n-1}=\{x\in \mathbb R^n: \sum_i x_i=0\}\subsetneq \mathbb R^n\;. \...
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1answer
57 views

Is $O(k)\times O(n-k)$ closed in $SO(n)$?

Let $O(m)$ denote the group of orthogonal matrices under multiplication, and let $SO(m)$ be the special orthogonal group over $\mathbb{R}$. Let \begin{equation*} (O(k)\times O(n-k))\cap SO(n):=\left\{...
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1answer
86 views

Why is $O(n)$ compact?

I'm not sure why $O(n)\subseteq R^{n\times n}$, the set of real orthogonal matrices of order $n$, is compact. I can see why it's bounded, but I still fail to understand why this would be a closed set. ...
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1answer
255 views

Find the orthogonal projection of the vector $(2, 3, 4)^t \in \mathbb{R^3}$ onto the $xy$ plane

Find the orthogonal projection of the vector $(2, 3, 4)^t \in \mathbb{R^3}$ onto the $xy$ plane where the symmetric bilinear product is given by the matrix \begin{pmatrix} 1 & 1 & 0 \\ ...
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2answers
62 views

Are all three matrices in Singular Value Decomposition orthornormal?

For Singular Value Decomposition (SVD) M = USV' I know that U and V are orthogonal matrices, but does S the diagonal matrix also have to be orthogonal? I know ...
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2answers
179 views

Sum of squared dot products of a unit vector with columns of an orthogonal matrix

Say we have an orthogonal matrix $\ U=[u_1 \ u_2 \ \cdots \ u_n]$, where $u_i, \ i={1,\ldots,n}$, is the ith columnn of $U$. If I took the dot product of a unit vector $x\in \mathbb{R}^n$ with each of ...