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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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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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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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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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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 ...
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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 ...
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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 ...
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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 ...
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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
120 views

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

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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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} ...
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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 ...
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1answer
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Is the orthogonal polar factor the unique retraction $\operatorname{GL}_n^+ \to \operatorname{SO}_n$?

$\newcommand{\psym}{\text{Psym}_n}$ $\newcommand{\sym}{\text{sym}}$ $\newcommand{\Sym}{\operatorname{Sym}}$ $\newcommand{\Skew}{\operatorname{Skew}}$ $\renewcommand{\skew}{\operatorname{skew}}$ $\...
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2answers
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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2answers
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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 $...
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1answer
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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 ...
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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 ...
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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 ...
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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 ...
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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 $G_{\mathbb{C}} = \mathrm{SO}(n,\mathbb{C})$ ...
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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 $U(n)$?
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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. ...
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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 $$...
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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 $\phi(\text{SO}(2))\le \text{SO}(n)$?
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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 ...
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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,\...
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1answer
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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 $E[\cdot]$...
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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, \...
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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 \\...
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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: $$ A = \begin{bmatrix} 1 & -1 & -7 \\ 1 & 3 & -1 \\ 2 & -1 & k \end{bmatrix} $$ ...
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2answers
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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 ...
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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>=...
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0answers
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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. ...
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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 ...
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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 ...
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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, $A = \operatorname{inverse}(A) = A^T$, ...
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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 ...
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3answers
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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 ...
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0answers
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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_{...
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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 $$A=(\cos\langle\alpha_i,\epsilon_j\rangle)_{n\times n}$$ $...
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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. $$Q = \text{arg}\min_{M \in O(n)} ||A - M||_{F}^2$$ ...
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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\...
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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 ...
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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 $\Pi \{ y_1,...,y_k\}$ ...
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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, $Q \in R^{m \times m}$
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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 ...
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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) = ...