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.
0
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1answer
44 views
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})$$(M^2)$$...
1
vote
1answer
21 views
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/\...
3
votes
1answer
28 views
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 ...
0
votes
0answers
18 views
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,...
2
votes
1answer
27 views
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?
1
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0answers
29 views
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, ...
0
votes
1answer
36 views
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}^...
0
votes
0answers
17 views
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 $2\times2$ ...
0
votes
2answers
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.
1
vote
2answers
49 views
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 ...
1
vote
0answers
32 views
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,...
1
vote
2answers
33 views
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 ...
2
votes
0answers
41 views
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 ...
-2
votes
0answers
38 views
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 ...
0
votes
1answer
32 views
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\...
0
votes
1answer
38 views
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!
4
votes
2answers
71 views
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 ...
0
votes
3answers
39 views
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|...
0
votes
0answers
22 views
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 ...
2
votes
2answers
41 views
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 = ...
0
votes
0answers
10 views
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 ...
0
votes
2answers
52 views
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 ...
1
vote
0answers
28 views
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
$$...
0
votes
2answers
39 views
Orthogonal matrix multiplied by diagonal matrix multiplied by transpose of the orthogonal matrix
Suppose tall matrix $A$ is $n \times k$ and that its columns are orthogonal, i.e., $A' A = I_k$. Suppose further that diagonal $M$ is $n \times n$ and has either $1$ or $0$ on its main diagonal. ...
0
votes
0answers
35 views
Proof based on orthogonal matrix
I'm hoping to show that $|Q| = +1$ or $-1$ if $Q$ is a $p \times p$ orthogonal matrix.
Since I know that $|QQ'| = |I|$ and $|Q||Q'| = |Q|^2$, then $|Q|^2 = |I|$
How should I approach this proof(...
0
votes
0answers
12 views
Why is the maximum value of an orthonormal dictionary of size $N \times N$ less than that of another having size $M \times M$, where M>N?
I was trying to compare the maximum value in DCT dictionary (representing an orthonormal dictionary) of size $N \times N$ to that of another DCT dictionary of size $M \times M$, where $M>N$. I ...
1
vote
2answers
30 views
Reducing the quadratic form
I'm trying to reduce the quadratic form $q(x_1, x_2, x_3, x_4) = x_1x_2 + x_1x_3 + x_1x_4 + x_2x_4$ into a quadratic form of the form $q = λ_1y_1^2 + λ_2y_2^2 + ··· + λ_ry_r^2$ for some real numbers $...
4
votes
3answers
58 views
When does $P\text{SO}P^{-1} \subseteq \text{SO}$?
Let $P \in \text{GL}_n^{+}(\mathbb {R})$. Suppose that $P\cdot \text{SO}(n)\cdot P^{-1} \subseteq \text{SO}(n)$.
Is it true that $P \in \lambda \text{SO}(n)$ for some $\lambda \in \mathbb{R}$?
I ...
0
votes
0answers
14 views
Index notation - notation for the inverse change of basis matrix with a hermitian metric
So, when we have any symmetic bilinear form $g = g_{ij} \epsilon^i \otimes \epsilon^j$, we can write $(A^{-1})^\mu{}_i = A_{\space i}{}^\mu$. This is one of the most beautiful things that index ...
0
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0answers
32 views
Does there exist any non-trivial linear relation on the components of elements of $O(2,1)$?
Consider as an example
$$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_{...
1
vote
1answer
30 views
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$, $\...
1
vote
2answers
61 views
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 ...
3
votes
2answers
89 views
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 ...
0
votes
0answers
42 views
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 ...
3
votes
2answers
55 views
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 $...
0
votes
1answer
28 views
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 ...
1
vote
1answer
27 views
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 ...
0
votes
0answers
27 views
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$ ...
0
votes
1answer
28 views
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$?
...
0
votes
1answer
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}$
$\...
2
votes
2answers
34 views
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}...
2
votes
2answers
29 views
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{...
2
votes
1answer
70 views
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}...
0
votes
1answer
76 views
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 ...
0
votes
2answers
69 views
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
...
1
vote
1answer
32 views
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 ...
1
vote
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 ...
7
votes
2answers
117 views
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 ...
4
votes
1answer
137 views
Is the orthogonal polar factor the unique submersion satisfying an orthogonality relation?
$\newcommand{\psym}{\text{Psym}_n}$
$\newcommand{\sym}{\text{sym}}$
$\newcommand{\Sym}{\operatorname{Sym}}$
$\newcommand{\Skew}{\operatorname{Skew}}$
$\renewcommand{\skew}{\operatorname{skew}}$
$\...
2
votes
2answers
44 views
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}
...
