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43
votes
4answers
2k 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
1answer
217 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
7answers
633 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
3answers
269 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
2answers
103 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
0answers
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 $O$,...
4
votes
1answer
132 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
1answer
139 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,
1
vote
0answers
100 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. ...
9
votes
2answers
638 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
1answer
125 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
1answer
280 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
1answer
149 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
1answer
249 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
0answers
190 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 ...
14
votes
2answers
286 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
1answer
94 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
2answers
212 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
1answer
164 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
1answer
249 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
3answers
168 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 \...
3
votes
2answers
329 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
2answers
86 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
2answers
832 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 ...
3
votes
2answers
511 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
1answer
134 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
2answers
1k 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 ...
2
votes
2answers
161 views

On $XX'=I$ such that $AX=XB$ is true when $A,B\in\{0,1\}^{n\times n}$

Given real symmetric matrices $A,B\in\{0,1\}^{n\times n}$ is it true that $$AX=XB$$ has a solution of form $X$ a permutation matrix iff a solution with $XX'=I$ exists? We are over reals. It is clear ...
6
votes
3answers
200 views

On $XX'=I$ such that $AX=XB$ is true

Given list of symmetric matrices $\{A_i,B_i\}_{i=1}^r\in\Bbb R^{n\times n}$ where $r\in\Bbb N$ is arbitrary what is a good description of collection of $X\in\Bbb R^{n\times n}$ such that $$XX'=I$$ $$...
7
votes
2answers
735 views

Geodesics on SO(3)

I have two 3D rotations about the origin, represented as $3 \times 3$ orthogonal matrices $M_1$ and $M_2$ (specified by numerical entries), and I would like to interpolate (and compute) a continuous ...
17
votes
0answers
406 views

Need explicit formula for certain “$q$-numbers” involving gcd's

The question is motivated by yet another possible approach to a combinatorial problem formulated previously in "Special" meanders. I'm not giving details of the connection as I believe the ...
19
votes
4answers
759 views

How much redundancy resides in an $n \times n$ orthogonal matrix?

Suppose one has an $n \times n$ orthogonal matrix $M$: $$ \left( \begin{array}{ccc} 0.239326 & 0.846726 & 0.475161 \\ 0.768893 & 0.13356 & -0.625272 \\ 0.592897 & -0....
1
vote
1answer
137 views

Dense symmetric unitary integer matrix?

Can someone give me a nontrivial example of a symmetric unitary integer matrix? I'm looking for something as dense as possible (i.e., not too many 0's); 5 <= size <= 8 would be ideal.
1
vote
0answers
92 views

MInors related problem [closed]

A matrix $A$ has $m$ rows and $n$ colums, such that $m \leq n$. We know that each row of $A$ has the norm $1$ (the norm of an element $x=(x_1,x_2,...,x_n) \in \mathbb{R}^n$ is $||x||=\sqrt{x_1^2+x_2^2+...
0
votes
0answers
134 views

Applying a linear operator to a basis set following SVD orthonormalization

Define $\Phi$ as an $N$x$N$ dense, symmetric matrix, who's columns represent a set of $N$ non-orthogonal bases. My intention is to: decompose $\Phi$ via SVD: $U \Lambda V^T = \Phi$ to create it's ...
4
votes
1answer
485 views

orthogonal group in characteristic 2

Let $O(2,\mathbb{Z}_2)$ be the orthogonal group of order two matrices. On $\mathbb{Z}_2$ there should exist just one odd quadratic form, hence the stabilizer subgroup $O^-$ of an odd quadratic for ...
2
votes
1answer
164 views

Existence of parametrizations of rational orthogonal matrices

I suppose that there are formulas which parametrize all the orthogonal matrices with rational coefficients. Does anyone know anything about it? And what are some publications that discuss this? Thanks....
0
votes
2answers
701 views

Is it possible to obtain the vectors orthogonal to a given one by orthogonal transformations?

Hello, everyone! Supposing that there is a unit vector in $n$-dimensional real space $\mathbf{x}_1\in\mathbb{R}^n$, I want to get a group of $n-1$ vectors to form an orthogonal basis with $\mathbf{x}...
1
vote
0answers
161 views

Ring-theoretic version of a matrix problem

Problem #17 in Zhan's survey of open problems in matrix theory is the Li-Poon problem on writing a square real matrix as the linear combination of $k$ orthogonal matrices. They proved that it is ...
4
votes
4answers
2k views

Parametrization of O(3)

Is there a simple way to parametrize the orthogonal group O(3) of 3 by 3 orthogonal matrices?