# Questions tagged [orthogonal-matrices]

Tag info removed....

63
questions

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votes

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60 views

### Maximize function on rotation matrices [closed]

Let $A$ be a fixed 3-by-3 matrix and $Q$ be a rotation matrix whose yaw, pitch, and roll angles are $\phi\in[0,\pi]$, $\theta\in[0,\pi]$, and $\psi\in[0,\pi/2]$, respectively:
\begin{equation}
Q=
\...

**2**

votes

**1**answer

71 views

### Characterization of extrinsic distance prevserving embedding (see the definition given!) from low dimensional Euclidean spaces to high dimensions

P.S. I asked the question on MSE more than a week ago, but didn't get any desired answer, so asking here.
Let $m < n \in \mathbb{N}$. Let us equip $\mathbb{R}^m, \mathbb{R}^n $ with their ...

**3**

votes

**1**answer

130 views

### LU decomposition for orthogonal or unitary matrices?

Is there any references on LU decomposition for orthogonal or unitary matrices?
It seems to me that the diagonal entries of $U$ has some nice structure regarding to the Euler angles of the original ...

**24**

votes

**2**answers

664 views

### Simple conjecture about rational orthogonal matrices and lattices

The following conjecture grew out of thinking about topological phases of matter. Despite being very elementary to state, it has evaded proof both by me and by everyone I've asked so far. The ...

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vote

**0**answers

140 views

### How to project a matrix to a unitary matrix?

Given a nonzero vector $v \in \mathbb{R}^n$, we all know that it's projection onto the unit $\ell_2$ ball is just $\frac{v}{\|v\|}$. Let $X$ be some nonzero $n \times n$ matrix. What is the projection ...

**1**

vote

**1**answer

103 views

### Minimize matrix norm over the unitary matrices

Suppose $C_1$ and $C_2$ are some fixed $n \times n$ matrices. Define the norm $\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$. What is $\min_U \|C_1 U C_2 \|$? Here $U$ ranges over the $n \times n$ unitary ...

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votes

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28 views

### Condition on the point cloud matrix making the points “generic” in the uniform sense

For a matrix $X\in\mathbb{R}^{d\times n}$, what condition can I impose on $X$ to make the collection of its columns generic in the sense that they look like the result of uniformly sampling a convex ...

**5**

votes

**1**answer

91 views

### How can I solve an orthogonal-constrained Sylvester equation?

I am currently facing a Sylvester equation
$AX+XB = C$
where $A$, $B$, $C$ are all symmetric and a special constraint here is that $X$ should be orthogonal. The Sylvester equation itself may not ...

**3**

votes

**1**answer

104 views

### Parametrizing quotient of matrices by the orthogonal group

I am trying to parametrize the collection of $d\times m$ real matrices quotient $d\times d$ orthogonal matrices. Formally, define $\sim$ on $\mathbb{R}^{d\times m}$ by $X\sim Y$ if there exists an ...

**6**

votes

**1**answer

246 views

### Can we choose smoothly the singular vectors of a matrix?

$\newcommand{\GLm}{\text{GL}_n^-}$Let $A$ be a real $n \times n$ matrix with non-positive determinant. Suppose that the smallest singular value of $A$ is strictly smaller than all the others (it has ...

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votes

**1**answer

118 views

### Upper bound on the sectional curvature of the orthogonal group

Consider the orthogonal group $O(n)$ as a Riemannian manifold endowed with the usual (bi-invariant) metric $\langle P, Q \rangle_A = \textrm{Tr}\ P^\top Q$ for tangent vectors $P, Q$, with $$T_A O(n) =...

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32 views

### Min-sum-max norm optimization with orthogonality constraint and matrix regularization [repost]

Disclaimer: this is a repost from https://math.stackexchange.com/q/3376158/443030, since the question may be a bit too elaborated.
Let $S = \{s_1,\cdots,s_N\}\subset\mathbb{R}^n$ be a finite set of ...

**4**

votes

**1**answer

92 views

### Distribution of Submatrix of Orthogonal Matrix

Let $O$ be a matrix sampled from the Haar measure on $O(n)$. Let $X$ be the upperleft $k\times k$ submatrix of $O$.
In a physics research project I am interested in the distribution of $X$, say $\rho(...

**2**

votes

**1**answer

226 views

### Bounding the Frobenius norm of orthogonalised matrices

Context: I am trying to show the convergence of an optimization method which includes orthogonalization in the update step.
Problem: Let's say I have real matrices $A, B \in \mathcal{R}^{n xm}$. If ...

**6**

votes

**3**answers

505 views

### Real orthogonal and sign [closed]

I came across the following conjecture, reading a recent paper in the Monthly, an orthogonal matrix of order $n\neq 0 \pmod 4$ has a nonnegative (up to a scalar) row vector.
It should be straight in ...

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votes

**2**answers

149 views

### Parametrising a sparse orthogonal matrix

I need to find a way to parametrise a matrix that is both sparse (to some degree) and orthogonal, i.e., I am looking for a parametrisation that describes $A \in \mathbb{R}^{n\times m}$ such that $AA^𝑇...

**4**

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**0**answers

367 views

### Orthogonal similarity of adjacency matrices of graphs which are cospectral and have a common equitable partition

Let $G$ and $H$ be two undirected graphs of the same order (i.e., they have the same number of vertices). Denote by $A_G$ and $A_H$ the corresponding adjacency matrices. Furthermore, denote by $\bar G$...

**3**

votes

**1**answer

164 views

### Generalization of Jordan's Lemma $A^2=B^2=I$ can be 2-block diagonalized

One of Jordan's lemma states that if two orthogonal matrices $A,B$ are such that $A^2=B^2=I$, then they can be co-diagonalized by block of size 2.
(the proof is easy, consider $x$ an eigenvector of $A+...

**11**

votes

**1**answer

436 views

### Is there a “formula” for the point in $\text{SO}(n)$ which is closest to a given matrix?

$\newcommand{\Sig}{\Sigma}$
$\newcommand{\dist}{\operatorname{dist}}$
$\newcommand{\distSO}[1]{\dist(#1,\SO)}$
$\newcommand{\distO}[1]{\text{dist}(#1,\On)}$
$\newcommand{\tildistSO}[1]{\operatorname{...

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83 views

### Infinitesimal matrix rotation towards orthogonality

TLDR; I am trying to prove the existence of an infinitesimal rotation which always moves a matrix "closer" to being orthogonal.
Setting
In this setting, we have a matrix $W \in \mathbb{R}^{n \times ...

**8**

votes

**1**answer

385 views

### A question about special linear group

Is there any way to find all matrices $G \in SL(n,\mathbb Z)$ such that there exists a matrix $A \in GL(n,\mathbb R)$ satisfying
$$
AGA^{-1} \in SO(n,\mathbb R)?
$$

**3**

votes

**0**answers

79 views

### How to find the best similarity transformation between two symmetric matrices $\mathbf{A}$ and $\mathbf{B}$? [duplicate]

Suppose I have two matrices $\mathbf{A}\in\mathbb{R}^{n\times n}$ and $\mathbf{B}\in\mathbb{R}^{n\times n}$. I want to know what's the best similarity transformation between these two matrices when we ...

**2**

votes

**0**answers

68 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 ...

**51**

votes

**4**answers

3k views

### When can one continuously prescribe a unit vector orthogonal to a given orthonormal system?

Let $1 \leq k < n$ be natural numbers. Given orthonormal vectors $u_1,\dots,u_k$ in ${\bf R}^n$, one can always find an additional unit vector $v \in {\bf R}^n$ that is orthogonal to the preceding ...

**5**

votes

**1**answer

304 views

### Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)

It's well known that the sectional curvatures of a Lie group, endowed with a left-invariant metric have a nice closed-form formula $k(X,Y) = \frac{1}{4} \|[X Y]\|^2$.
I'm wondering if the following (...

**13**

votes

**7**answers

893 views

### Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?

(Disclaimer : I know very well that $SO(N)$ has a Lie algebra of dimension $N(N-1)/2$ etc. This absolutely not the point of my question.)
To make my problem more understandable, I start with the ...

**2**

votes

**3**answers

308 views

### Inverse of matrix $D + ADA^T$

Let $D$ be an arbitrary diagonal matrix and let $A$ be an orthogonal matrix ($A'A = AA' = I$). How to compute the following matrix inverse efficiently?
$$(D + ADA^T)^{-1}$$
Hints or references are ...

**4**

votes

**2**answers

231 views

### Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix

Given two $N \times N$ symmetric matrices $A, B$, where $A$ is positive semidefinite while $B$ is not positive semidefinite. I am interested in solving unitary constrained trace maximization problem:
...

**2**

votes

**0**answers

70 views

### Characterizing a subclass of row-orthogonal matrices

Let $O\in\mathbb{R}^{n\times m}$, $m>n$, be such that $O O^\top =I_n$. (Here $\bullet^\top$ denotes transposition and $I_n$ the $n\times n$ identity matrix.) Consider the following partition of $O$,...

**5**

votes

**1**answer

260 views

### Positive definite matrices diagonalised by orthogonal matrices that are also involutions

Let $A$ be a positive definite matrix. Then, $A$ is diagonalized by an orthogonal matrix $P$.
I want to know when this matrix is also an involution, i.e., $P^2 = I$.
If there is any ...

**2**

votes

**1**answer

144 views

### Finding a similarities and differences of sent of matrices

Suppose we have a set of rank deficient covariance matrices. How can I know the similarities and differences between those set of matrices?
Regards,

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**0**answers

138 views

### A question about permutation matrices

This question is trying to abstract out in a self-contained way the point that is probably being made in page 6 of this paper, https://arxiv.org/pdf/1604.03544.pdf and why Theorem 4.1 there works.
...

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votes

**2**answers

1k views

### Is there a standard name for (non-square) matrices with orthonormal columns?

One encounters often in numerics non-square matrices with orthonormal columns, i.e., $U\in\mathbb{R}^{m\times n}$, with $m > n$, such that $U^TU=I$ (but, clearly, $UU^T \neq I$).
Is there a name ...

**2**

votes

**1**answer

196 views

### What's the best orthonormal matrix to align two matrices in the operator norm sense?

Let $A,B \in R^{n\times r}$ with $A^\top B $ invertible. It is known that
\begin{equation}
UV^\top :=\arg\min_{R \in \mathcal{O}^{r\times r}}\|AR-B\|_\mathrm{F},
\end{equation}
where $USV^\top$ is ...

**2**

votes

**1**answer

519 views

### maximum of orthogonal vectors

$$v_1=(x_1,x_2,x_3\cdot\cdot\cdot,x_n)$$is such a vector. By changing its signs and positions of each component $x_i$, we can get different vectors. When n is odd, it's impossible for any of ...

**4**

votes

**1**answer

163 views

### What is the term for this type of matrix?

Is there an established term for the following type of square matrices?
$\begin{pmatrix}
c & c & c & c & \cdots & c & c \\
c & a & b & b & \cdots & b & ...

**2**

votes

**1**answer

291 views

### What is the term for a matrix whose columns are orthogonal?

What is the term for a matrix whose columns are mutually orthogonal, but not necessarily othonormal?
I can't name such a matrix "orthogonal" because that would imply that all columns are unit vectors....

**3**

votes

**0**answers

240 views

### Singular value decomposition of a low rank weak diagonally dominant M-matrix. When is the unitary polar matrix positive semi-definite?

Let $A$ be an $n \times n$, non-symmetric, real, weak diagonally dominant M-Matrix. Its diagonal is strictly positive, its off-diagonal is negative or zero and all its columns sum to zero. $A$ has ...

**15**

votes

**2**answers

333 views

### matrix inequality with orthogonal matrices

I would like to know if for $A,B\in SO(3)$ the inequality
$$
\|AB-BA\|_F\leq \|A-I\|_F\|B-I\|_F
$$
holds, where $\|\cdot\|_F$ denotes the Frobenius norm and $I$ the identity matrix. Using the identity
...

**-1**

votes

**1**answer

116 views

### Orthogonal polynomials of the second kind

Let $L: \mathbb{R}[x] \rightarrow \mathbb{R}$ be a positive definite linear functional and let that $\{s_n\}$ be a positive semi-definite sequence such that $L(x^n)= s_n, n\ge 0.$ Given a positive ...

**6**

votes

**2**answers

220 views

### Bounding the non-multiplicativity of isometric projection

Every $A \in \text{GL}_n(\mathbb{R})$ has a unique Polar decomposition:
$A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$.
In particular the orthogonal factor is given by $$O_A=A(\...

**5**

votes

**1**answer

186 views

### When does isometric projection respect multiplication?

Every $A \in \text{GL}_n(\mathbb{R})$ has a unique orthogonal polar factor $O_A=A(\sqrt{A^TA})^{-1}$,
( $A=O_AP_A$, $O \in \operatorname{O}_n, P \in \operatorname{Psym}_n$see Polar decomposition).
...

**0**

votes

**1**answer

392 views

### A question on orthogonal matrix [closed]

Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.

**6**

votes

**3**answers

191 views

### What is special in dimension $2$ (When characterizing isometries using the cofactor matrix)?

Let $A$ be a real $n \times n$ matrix. Denote by $\operatorname{cof} A$ The cofactor matrix of $A$. By definition, $A (\operatorname{cof} A)^T=\det A \cdot I$.
Thus, it is immediate that $A \in \...

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votes

**2**answers

523 views

### Partitioning an orthogonal matrix into full rank square submatrices

Let $U$ be an $n \times n$ orthogonal matrix. Given an arbitrary partition ${\mathcal P}_c=\{y_1,y_2,\ldots,y_k\}$ of the columns of
$U$, does there always exist a corresponding partition ${\mathcal ...

**0**

votes

**2**answers

102 views

### Functions with scalar times orthogonal Jacobian [duplicate]

I am interested in understanding functions $f:\mathbb{R}^d \rightarrow \mathbb{R}^d $ whose Jacobian at every point $x \in \mathbb{R}^d$ is a scalar times an orthogonal matrix.
I've seen a similar ...

**4**

votes

**2**answers

1k views

### Why decompose a function with eigenvectors of Laplace operator? [closed]

On periodic domain, people always use Fourier basis, which eigenvectors of Laplace operator. On sphere, people use spherical harmonics, which also are eigenvectors of Laplace operator. In applied ...

**4**

votes

**2**answers

907 views

### functions with orthogonal Jacobian

I'm working on a model that would require to use vectorial functions of $\mathbb{R}^n \rightarrow \mathbb{R}^n$, such that $\forall x, y \in \mathbb{R}^n$, $\lVert \frac{df(x)}{dx}(y) \lVert_2 = \...

**2**

votes

**1**answer

147 views

### Heuristics for counting degrees of freedom

I have recently learned about the representation theorem for isotropic,
linear operators, which says the following:
Defintion:
Let $M_n$ be the vector space of $n \times n$ real matrices. We say a ...

**7**

votes

**2**answers

2k views

### The space of positive definite orthogonal matrices

The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite.
$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal.
and the ...