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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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Explain whether this matrix is symmetric or not?

I have a matrix $M$ and another $N$. $N$ is an orthogonal (orthogonal => $N^{T} = N^{-1})$ r x r matrix and $M$ is an r x r skew symmetric matrix (skew syemmtric => $M^{T} = -M$). Is $(N^{-1})$$(M^2)$$...
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The fundamental groups of 3-dimensional spherical space forms

Let $S^3/\Gamma_i\,(i=1,2)$ be a $3$-dimensional spherical space form, where $\Gamma_i \subset SO(4)$ is a finite subgroup acting freely on $S^3$. If $S^3/\Gamma_1$ is homotopy equivalent to $S^3/\...
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Are two isomorphic finite subgroups of $SO(4)$ conjugate?

Let $A,B$ be two finite subgroups of $SO(4)$ such that $A$ and $B$ are isomorphic as abstract groups. Can we find a $g \in SO(4)$ such that $$ gAg^{-1}=B? $$ If it is the case, does the same ...
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Expected value for the eigenvalue of random orthogonal matrix

I am interested in finding the expected value of the product of two eigenvalue with respected to the GOE ensemble. So I have a random $n\times n$ matrix with all off diagonal elements $\mathcal{N}(0,...
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Homotopy groups of split orthogonal group

What are the homotopy groups of $O(d,d)$? Is it possible to compute them somwhow by using the fact that $O(d)\times O(d)$ is the maximal compact subgroup, and the homotopy groups of $O(d)$ are known?
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Are the invariants related in the characteristic equation of an orthogonal matrix (3x3 matrix)?

The characteristic equation of matrix A is $$\lambda ^3 - I_1\lambda^2 + I_2\lambda-I_3 = 0 $$ For orthogonal matrix $$I_3 = det(A) = \pm1$$ $$I_1 = tr(A)$$ Taking examples of orthogonal matrices, ...
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Problems with determining the orthonormal basis regarding an inner product $B$

Can somebody tell me where I made a mistake? How would you approach such an exercise? Was my way too complicated? $ B:\mathbb{R}^3\times \mathbb{R}^3\to \mathbb{R},\quad B(x,y)=\sum \limits_{i\neq j}^...
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Counting $2\times2$ Orthogonal matrices over the ring $\Bbb{Z}_p[i]$. [duplicate]

Our research is about counting the number of orthogonal matrices over the ring of Gaussian integers modulo $p$. A matrix $A$ is said to be orthogonal if $AA^T=I$. My question is how many $2\times2$ ...
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39 views

For an orthogonal matrix $Q$, prove $\operatorname{cond}(Q)=1$

Given an orthogonal matrix $Q$, prove $$\|Q\|_2\cdot \|Q^{-1}\|_2=1$$ I succeed to solve it with eigenvalues but I'm looking for an easier way.
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Prove that special orthogonal matrix from $SO(n, \mathbb{R})$ has $\frac{(n-1)n}{2}$ independent parameters

Show that special orthogonal matrix from $SO(n, \mathbb{R})$ has $\frac{(n-1)n}{2}$ independent real parameters. I assume that this will be related to Euler angles somehow or specifically to its ...
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What does the “standard basis” of $O(1,n)$ mean?

Let $O(1,n)$ be the orthogonal group of the quadratic form $b(x)=-x_0^2+\sum\limits_{i=1}^n x_i^2$. In other words, $$O(1,n)=\{T\in GL(n+1,\Bbb{R}|b(Tx)=b(x),\forall x\in R^{n+1}\}$$ Elements in $O(1,...
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Distance of matrix to $\mbox{SO}(n)$ w.r.t. Frobenius norm

Given $A \in \mathbb R^{n\times n}$, I was told that $$\mbox{dist}(A, \mbox{SO}(n)) = \inf_{Q \in \mbox{SO}(n)} |A-Q| \overset ? = \inf_{Q \in \mbox{SO}(n)} | Q^\top A - \text{Id} |$$ where we use ...
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Complex-analytic isometry of hyperbolic ball and half-space

I'm trying to prove that the $n$-dimensional Poincaré ball and half-space models of hyperbolic space are isometric. Here the Poincaré ball $\mathbb B^n_R$ ($n$-ball of radius $R$) has the Riemannian ...
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eigen vectors of 3x3 rotation matrix properties

First let me add to the questions some statements which I know for fact, and which is necessary for the background. 1> the eigen vectors of a rotation matrix given a rotation axis vector is ...
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Orthogonal matrix whose first column is given

What would be an orthogonal matrix whose first column is $\underline{x} = \begin{vmatrix} -1\\ \underline{y}\\ \end{vmatrix}$, where $\underline{y} \in {\rm I\!R}^{n-1} $, $\underline{x} \in {\rm I\...
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Matrix multiplication commutative property

Let A=(I-S)(I+S), can it be written as A=(I+S)(I-S) where I is identity matrix and S is n rowed real skew symmetric matrix? I have a question and in solution they wrote it. Thanks!
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Every real matrix $A$ is the linear combination of $4$ orthogonal matrices

Question: Prove that every matrix $A\in M_n(\mathbb R)$ is the linear combination of $4$ orthogonal matrices $X, Y, Z, W$ , i.e. $A=aX+bY+cZ+dW$ for some $a,b,c,d\in\mathbb R$. This problem is ...
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Question about a proof that the eigenvalues of an $n \times n$ orthogonal matrix are $\pm 1$

Suppose A: $n \times n$ orthogonal matrix, $\lambda$ is an eigenvalue of A and $x$ is corresponding eigenvector. We know that $Ax = \lambda x$ Then $(Ax)^T (Ax) = x^T A^T Ax = (Ax) \cdot (Ax) = |Ax|...
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Haar measure from axis-angle representation of $SO(3)$

My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure ...
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Matrices in $\operatorname{O}(n) \setminus \operatorname{SO}(n)$

Can any matrix $M \in \operatorname{O}(n) \setminus \operatorname{SO}(n)$ be written as $I_n - uu^T$ where $I_n \in \mathbb{R}^{n \times n}$ is the identity matrix and $u \in \mathbb{R}^n$, $||u||_2 = ...
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Householder matrices for thin QR updating

Suppose you have a QR decomposition of the form: $X=QR$ (Where $X$ is an arbitrary matrix of size $(n \times p)$, $Q$ is an orthogonal matrix of size $(n \times n)$ and $R$ is an upper trapezoidal ...
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Does 3x4 matrix have an Inverse? Why? [duplicate]

I saw this question somewhere and made me think do 3x4 matrices have an inverse, as I previously that that only square matrices have an inverse. If non-square matrices have an inverse, especially if ...
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Minimum singular value of sum of rotations

Consider orthogonal matrices $Q_1, Q_2 \in \mathbb{R}^{d \times d}$, where both matrices are proper rotations of angle $\theta_1, \theta_2$ around different axes. Now, consider the symmetric matrix $$...
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Orthogonal matrix multiplied by diagonal matrix multiplied by transpose of the orthogonal matrix

Suppose tall matrix $A$ is $n \times k$ and that its columns are orthogonal, i.e., $A' A = I_k$. Suppose further that diagonal $M$ is $n \times n$ and has either $1$ or $0$ on its main diagonal. ...
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Proof based on orthogonal matrix

I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix. Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$ How should I approach this proof(...
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Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?

I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
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2answers
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Reducing the quadratic form

I'm trying to reduce the quadratic form $q(x_1, x_2, x_3, x_4) = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_4$ into a quadratic form of the form $q = λ_1y_1^2 + λ_2y_2^2 + ··· + λ_ry_r^2$ for some real numbers $...
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When does $P\text{SO}P^{-1} \subseteq \text{SO}$?

Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$. Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$? I ...
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Index notation - notation for the inverse change of basis matrix with a hermitian metric

So, when we have any symmetic bilinear form $g = g_{ij} \epsilon^i \otimes \epsilon^j$, we can write $(A^{-1})^\mu{}_i = A_{\space i}{}^\mu$. This is one of the most beautiful things that index ...
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Does there exist any non-trivial linear relation on the components of elements of $O(2,1)$?

Consider as an example $$O(2)=\left\{\begin{pmatrix}\cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}\middle|\ \ \theta\in \Bbb R\right\},$$ clearly for $g\in O(2)$ one has $g_{...
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How to prove that $\langle P,A^2 \rangle \le 0 $ for every positive $P$ and skew-symmetric $A$?

I have stumbled upon the following claim, and I wonder if it has a simple proof: Let $P$ be a real $n \times n$ symmetric positive definite matrix. Then for every real skew-symmetric matrix $A$, $\...
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Prove that all complex eigenvalues of the operators of a unitary or orthogonal representation have modulus 1.

The question is given below: Let $T$ be an orthogonal or unitary representation of the group $G$. Prove that all complex eigenvalues of the operators $T(g)$, $g \in G$ have modulus one. But I do ...
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Orthogonal matrix with single $0$ entry

To my surprise there is an orthogonal matrix dimension $3 \times 3$ with a single $0$ entry as it was shown in this answer. Moreover it was possible to identify the pattern for matrix entries ...
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Orthogonal block matrix made of (signed) permutation matrices

Let $\{P_1, \cdots, P_n\}$ be $n$ permutation matrices with size $n \times n$. I'd like to build a $n^2 \times n^2$ matrix $P$ such that $P^\top P=P P^\top$ is a multiple of the identity, and ...
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Number of possible zero entries in orthogonal matrices

It's easy to check that in an orthogonal matrix $Q$ dimension $2 \times 2$ if there is entry $0$ in the matrix then necessary one additional zero must be present and the total number of zeros is $...
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1answer
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Find $\text{card}(T_n( \mathbb R ) \cap O(n))$.

Let $T_n( \mathbb R )$ be the set of upper triangular matrices of size $n$. Let $O(n)$ be the set of general orthogonal matrices and $SO(n)$ the set of special orthogonal matrices. Find the cardinal ...
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How to find the near close aspect ratio of a billboard from a distance photo?

I have a photo of a billboard. I am trying to solve this problem of finding out the aspect ratio of this billboard. What is known to me is nothing more than this photo. Can someone help me with ...
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Band matrix conjugation relation between orthogonal matrices

Suppose I have two orthogonal matrices: $C_1,C_2$ of some dimension $n$. It is given that $C_2 = R_1*C_1*R_2$ where R1,R2 are orthogonal matrices as well. How can I solve for $R_1$ and $R_2$ ...
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Linear Algebra, orthogonal columns and length

Suppose A is a 3x3 matrix whose columns are orthogonal and the length (two-norm) of each column equals 4. Then what is $A^T*A$? ...
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36 views

Is there a closed-form formula for the derivative of the orthogonal polar factor of a matrix?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\newcommand{\SO}{\operatorname{SO}_n}$ $\...
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Orthogonal matrices only defined for standard inner product?

$\newcommand{\tp}[1]{#1^\mathrm{T}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\n}{\{1,\ldots,n\}} \newcommand{\siff}{\quad\Leftrightarrow\quad} \newcommand{\ijth}[2][\tp{Q}Q]{[#1]_{#2}} \newcommand{\K}...
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2answers
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For which $P,Q \in \text{SO}$ $T_P\text{SO}$ and $T_Q\text{SO}$ are parallel?

I am curious: For which $P,Q \in \text{SO}_n$ does $T_Q\text{SO}_n=T_P\text{SO}_n$ hold? This reduces to the question at the identity,i.e. for which $Q \in \text{SO}_n$, $T_Q\text{SO}_n=T_{Id}\text{...
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1answer
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Invariant subsets of orthogonal representation of symmetric 3 tensor.

Let $O(n)$ be the orthogonal (matrix) group. We consider the action of $O(n)$ on the space of symmetric $3$-tensor $S^{0,3}$, where the action is $$ A\mapsto O^* A , \ \ \ (O^*A)_{ijk}= A_{abc} O_{ai}...
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Do orthonormal matrices have a geometric interpretation or contextual importance? [closed]

I understand how certain matrices can be orthonormal and the conditions necessary for that. I don't understand its geometric relevance. For instance in a vector space the axis would be orthonormal and ...
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how to find the orthogonal projection of u onto v

I would love some help with a question I dont know how to answer. Let $ u=\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}^T,V= \begin{bmatrix} 1 & i \\ -i & 1 \\ 1 & 0 ...
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1answer
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Are all symplectic $(0,1)$-matrices lower/upper block-triangular?

Context. I don't expect this question is actually interesting, it just seems like a nice/fun exercise to get better acquainted with $Sp(2n)$ (and to celebrate the new year!). In this question ...
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1answer
52 views

For given column vectors $x$, $y$ how to find an orthogonal matrix $Q$ such that $Qx=y$

Problem: Given $x=[1,7,2,3,-1]^T $and $y=[-4,4,4,0,-4]^T$, find an orthogonal matrix $Q$ such that $Qx=y$. My attempt: I know the definition of an orthogonal matrix. By definition, if $Q$ is an ...
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Which matrices can be realized as second derivatives of orthogonal paths?

$\newcommand{\skew}{\operatorname{skew}}$ $\newcommand{\sym}{\operatorname{sym}}$ $\newcommand{\SO}{\operatorname{SO}_n}$ I am interested to know which real matrices $A \in M_n$ can be realized as ...
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1answer
137 views

Is the orthogonal polar factor the unique submersion satisfying an orthogonality relation?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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2answers
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How can we decompose the identity matrix given a set of orthonormal vectors?

Let $A$ be a positive semidefinite (P.S.D) matrix with distinct set of eigenvalues. since it is P.S.D its eigendecomposition is as follows for eigenpairs of $(\lambda_i,v_i)$ $$ A= \begin{bmatrix} ...