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.
2
votes
2answers
28 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
26 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{...
0
votes
0answers
12 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
72 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
43 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
25 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
48 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
103 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
120 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
41 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}
...
4
votes
1answer
105 views
Which matrices commute with $\operatorname{SO}_n$?
$\newcommand{\GLp}{\operatorname{GL}_n^+}$
$\newcommand{\SO}{\operatorname{SO}_n}$
Let $n>2$, and Let $A \in \GLp$ be an invertible real $n \times n$ matrix, which commutes with $\SO$.
Is it true ...
0
votes
1answer
24 views
Is the orthogonal polar factor the unique retraction $\operatorname{GL}_n^+ \to \operatorname{SO}_n$?
$\newcommand{\psym}{\text{Psym}_n}$
$\newcommand{\sym}{\text{sym}}$
$\newcommand{\Sym}{\operatorname{Sym}}$
$\newcommand{\Skew}{\operatorname{Skew}}$
$\renewcommand{\skew}{\operatorname{skew}}$
$\...
1
vote
2answers
29 views
If $A$ has orthonormal columns then $||Ax||^2_2 = ||x||^2_2$, why?
In the lecture notes we have a fact:
If $A$ has orthonormal columns then $||Ax||^2_2 = ||x||^2_2$
Why is it the case? What properties of matrix-vector multiplication should I know to reason about ...
0
votes
0answers
24 views
Basis of an Orthogonal Complement
Let $V$ be the vector space of all $2$ by $2$ matrices. Let $<M_1, M_2>$ $=$ $tr(M_1^TM_2)$ be an inner product defined on $V$. Let $A$ $=$ $\begin{pmatrix}1&1\\ 1&0\end{pmatrix}$ be one ...
0
votes
0answers
29 views
Orthogonal Matrices and Cosets (translates) of Linear Subspaces
Let $M_n(F_2)$ be the vector space of all $n\times n$ matrices over the finite field $F_2$. Let $O(n)\subset M_n(F_2)$ be the set of all orthogonal matrices and $W\subseteq O(n)$ be an affine subspace ...
0
votes
0answers
34 views
Minimizing Frobenius norm of the difference between $A$ and $P^\top B P$
I have two $n\times n$ real matrices $A$, $B$, not necessarily symmetric.
Is there any clever way to compute
$$
\min_P || A - P^{\top} B P ||_F
$$
where $P$ ranges over the group of orthogonal ...
0
votes
0answers
28 views
Parameterize Orthogonal Matrix where determinant is -1
Since orthogonal matrix is a disconnected topological space (can be separated into determinant 1 and determinant -1)
So I guess we can't parameterize both groups with single parameterization.
I have ...
2
votes
2answers
49 views
Orthogonal transformation with additional constraints
Let $A$ be an orthogonal matrix, i.e. $AA^{T}=\mathbb{I}$. It is given that $A$ satisfies an additional constraint, $AMA^{T}=PMP^{T}$, where $P$ is some permutation matrix and $M_{ij}=sgn(i-j)$. Can $...
0
votes
1answer
16 views
Show that $L_A$ acts on by orthogonal transformation and in particular rotation.
Let $A$ be a $3\times 3$ orthogonal matrix with determinant $=1$.
Let $v$ be an eigen vector corresponding to $1$ of $A$.Let $W=\text{span}\{v\}$.
Show that $L_A$ preserves $W^\perp$ and it acts ...
1
vote
1answer
26 views
Show that there is an orthogonal matrix $O$ such that $OA_1=A_2O$.
Let $A_1,A_2$ be two real $n \times n$ matrix. And suppose that they are two orthogonal and anti-symmetric matrices. Show that there is an orthogonal matrix $O$ such that $OA_1=A_2O$.
I have no idea ...
1
vote
1answer
20 views
Can a linear isometry always be expressed in terms of an orthogonal matrix?
Is the following true?
Let $S: \mathbb{R}^n \to \mathbb{R}^n$ be a linear transformation such that $||S(v)|| = ||v|| \ \text{for all} \ v \in \mathbb{R}^n$, where $||\cdot||$ denotes the Euclidean ...
0
votes
0answers
21 views
Property of $Q$ such that $\frac{x^TQx}{\|x\|^2} = \text{const}, \ \ \ \forall x\in \mathbb{R}^n$
I am curious about the following problem:
suppose $Q^TQ = I$, i.e., $Q$ is orthogonal
we want $$\frac{x^TQx}{\|x\|^2} = \text{const}, \ \ \ \forall x\in\mathbb{R}^n$$
My question is what ...
1
vote
1answer
49 views
Left cosets complex orthogonal group under real orthogonal group
In the lecture notes at http://www.math.ias.edu/QFT/fall/lect2.ps (page 2) there is a "standard" lemma:
In this lemma $G = \mathrm{SO}(n,\mathbb{R})$ and $G_{\mathbb{C}} = \mathrm{SO}(n,\mathbb{C})$ ...
2
votes
0answers
34 views
Higher dimensional version of “a product of two reflections is a rotation”
In higher dimensions, which orthogonal matrices are a product of two reflections? Is it all of $SO(n)$? In the complex case is it $U(n)$?
0
votes
1answer
22 views
How to determine if three vectors form a basis for a subspace?
This is a follow up question ( math.stackexchange.com/q/3018473); i'm interested in understanding some other part of the problem.
I have three vectors, v1, v2, v4, which are linearly independent.
...
3
votes
2answers
38 views
Can every orthogonal matrix be written as a product of Givens rotations?
I'd like to know whether every orthogonal matrix
$$ A \in \mathcal{O}_n(\mathbb{R})$$
can be written as a product of givens-rotations. I know that when we do QR-decomposition of matrix $A$ we get
$$...
2
votes
1answer
60 views
Do every two orthogonal matrices in $\text{SO}(n)$ lie in the same coset of $\text{SO}(2)$?
Let $A,B \in \text{SO}(n)$.
Does there exist a homomorphism of Lie groups $\phi:\text{SO}(2) \to \text{SO}(n)$, such that $A,B$ lie in the same coset of $\phi(\text{SO}(2))\le \text{SO}(n)$?
0
votes
0answers
24 views
Find eigenvalues, kernel and Image of an Orthogonal projection
Let $V$ be a Vector space with inner product and $U$ a subspace. Let $P$ be the orthogonal projection over $U$. Find eigenvalues, kernel and Image of $P$.
I know I have to consider the special cases ...
2
votes
1answer
58 views
Does orthogonal-invariance of a differential imply invariance of the function?
Let $U:\text{Hom}(\mathbb{R}^d,\mathbb{R}^d) \to \mathbb{R}$ be a smooth function .
If $U$ is orthogonally-invariant, i.e. $U(QA)=U(A)$ for every $Q \in \text{SO}(n),A \in \text{Hom}(\mathbb{R}^d,\...
0
votes
1answer
12 views
Joint Gaussian PDF Change of Coordinates
My textbook says the following:
Given a vector $\mathrm{\mathbf{x}}$ of random variables $x_i$ for $i = 1, \dots, N,$ with mean $\bar{\mathrm{\mathbf{x}}} = E[\mathrm{\mathbf{x}}]$, where $E[\cdot]$...
3
votes
3answers
157 views
Proving that a matrix is symmetric if it can be expressed as a spectral decomposition
If $\{u_1, \cdots, u_n\}$ is an orthonormal basis for $\mathbb{R}^n$, and if $A$ can be expressed as
$$A = c_1u_1u_1^T + \cdots + c_nu_nu_n^T$$
then $A$ is symmetric and has eigenvalues $c_1, \...
0
votes
0answers
29 views
Consider the plane P in R-3 given by x-y-2z=0
I found the matrix A whose columns are a basis for P, A=[1,-1,-2] (vertical form).
Using that I was able to find the projection matrix:
P=$\frac{-1}{2} \left( \begin{array}{cc} 1 & -1 & 2 \\...
0
votes
0answers
41 views
Find constant $k$ in matrix so that matrix $A$ is orthogonal
Problem
Find constant $k\in \mathbb{R}$ in matrix so that matrix $A$ is orthogonal when:
$$ A = \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix} $$
...
0
votes
2answers
34 views
Orthogonal Diagonalization of a $3$ by $3$ Matrix
$M$ $=$ $\begin{pmatrix}3&2&2\\ 2&3&2\\ 2&2&3\end{pmatrix}$. Diagonalize $M$ using an orthogonal matrix.
So I got that the eigenvalues for $M$ were $1$ and $7$. For the ...
0
votes
1answer
43 views
Find out if a operator is self-adjoint or orthogonal
Let $V$ be a $\mathbb{R}$ inner product space, and $B=\left \{v_1, v_2, v_3 \right \}$ basis of $V$, with $||v_i||=1$ $\forall i=1,2,3$, and $<v_1, v_2>=<v_1, v_3>=0$ and $<v_2, v_3>=...
1
vote
0answers
49 views
How to show that $SO(3)$ and $\mathbb{R}P^3$ are dffeomorphic?
Pardon me for repeating a question, but I am not able to justify one aspect of this proof that $SO(3)$ and $\mathbb{R}P^3$ are diffeomorphic, and I was hoping that someone could clarify it for me.
...
0
votes
0answers
19 views
Show that Y is isotropic in V iif, for some v in V, the random vectors Y-v and O(Y-v) have the same mean and variance
Problem:
Show that Y is isotropic in V iif, for some v in V, the random vectors Y-v and O(Y-v) have the same mean and variance for all orthogonal linear transformations O: V --> V
I understand that ...
0
votes
1answer
42 views
Can we write any unitary matrix as the exponential of a skew-symmetric complex matrix?
Following the article https://en.wikipedia.org/wiki/Skew-symmetric_matrix#Infinitesimal_rotations which states that a skew symmetric (or anti-symmetric) real matrix can be written as the exponential ...
0
votes
2answers
64 views
Proof of orthogonal and symmetric.
Given $x$ is an $n$ dimensional vector, if $A = I_n- (2/x^Tx)xx^T$, show that it is orthogonal and symmetric.
I know that if $A$ is orthogonal and symmetric, $A = \operatorname{inverse}(A) = A^T$, ...
2
votes
2answers
55 views
Is the solution to $A-O(A)=\tilde \Sigma$ unique?
Let $\tilde \Sigma=\text{diag}(\tilde \sigma_i)$ be a diagonal matrix, with $\tilde \sigma_i>0$. ($1 \le i \le n$).
Suppose that $A$ is a real invertible $n \times n$ matrix with positive ...
2
votes
3answers
61 views
Linear-algebra first course problem about orthogonal matrices
I am trying to demonstrate next assert about matrices:
$A$ is a matrix of $n$ order, with $n$ odd, that obeys $A A^T =I$ and $\det\, A=1$. Then $\det\,(A-I)=0$.
I have tried a number of things but ...
1
vote
0answers
39 views
Equivalence classes of orthogonal matrices
Consider $O(n)$, the Lie group of $n{\times}n$ orthogonal matrices.
Let $\pi$ be a given permutation of $(1,2,\ldots,n)$;
the elements of the corresponding permutation matrix are
\begin{eqnarray}
P_{...
0
votes
1answer
36 views
Prove matrix $A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$ is orthogonal
Assume $\{\alpha_1,\cdots\alpha_n\},\{\epsilon_1,\cdots,\epsilon_n\} $ are both orthonormal basis of Euclidean Space $V$. Consider the matrix $$A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$$ $...
5
votes
1answer
135 views
Nearest signed permutation matrix to a given matrix $A$
Let $A \in \mathbb{R}^{n\times n}$ be a square matrix and let $Q \in O(n)$ be the nearest orthogonal matrix to $A$ under the Frobenius norm, i.e.
$$Q = \text{arg}\min_{M \in O(n)} ||A - M||_{F}^2$$
...
0
votes
1answer
20 views
Project an orthogonal matrix onto the Birkhoff Polytope
It is known that the permutation matrices lie at the intersection of the orthogonal group $\mathbb{O}^N$ with the Birkhoff polytope $\mathbb{DS}^N$. It is also known that any non-negative matrix $X\in\...
1
vote
0answers
35 views
On a special decomposition of a $3\times 3$ matrix
Let $A\in\mathbb{R}^{3\times 3}$ be a diagonalizable matrix with strictly positive eigenvalues. (Note that $A$ is not required to be symmetric.)
Let $A_S$ be the symmetric part of $A$, that is
$$
A_S ...
0
votes
1answer
30 views
If vector $y_1 = (1,2,3)$ and $y_2 = (4,5,6)$, how to calculate orthogonal projector $\prod\{y_1\}$ onto subspace spaned by the vectors $y_1,y_2$?
If I have a vector $y_1 = (1,2,3)$ and $y_2 = (4,5,6)$, how to calculate $\prod\{y_{1}\}$ and $\prod\{(y_{i})_{1\leq i \leq2}\}$ according to the definition below?
denote $\Pi \{ y_1,...,y_k\}$ ...
0
votes
3answers
95 views
if $Q^TQ = I$, can we get $QQ^T=I$
if we only know $Q^TQ = I$, can we get $QQ^T=I$?
where $I$ is the identify matrix, $Q \in R^{m \times m}$
4
votes
2answers
77 views
$A^TA=B^TB$. Is $A=QB$ for some orthogonal $Q$?
Suppose that $A$ and $B$ are two real square matrices and $A^TA=B^TB$. Can we say that $A=QB$ for some orthogonal matrix $Q$?
If they are vectors we have $\|a\|^2=a^Ta=b^Tb=\|b\|^2$, so intuitively ...
0
votes
1answer
79 views
Classical Lie group quotient-ed by its maximal parabolic subgroup
Let $B$ is a nondegenerate symmetric bilinear form on $\mathbb{C}^n$ then the corresponding complex orthogonal group is $\{g : GL(n, \mathbb{C}): B(gx, gy) =(x,y) \}$
In particular we use $$B (x,y) = ...
