Skip to main content

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.

4 votes
1 answer
458 views

If the sum of each row and column of an orthogonal matrix equals 1, is it a permutation matrix?

A permutation matrix has a single $1$ in each row and column, with the rest of the entries in that row and column equal to zero. Question: If I have an $n\times n$ orthogonal matrix $\mathbf{Q}$ ...
user3433489's user avatar
3 votes
4 answers
391 views

Distance Between Subspaces

Let $P_1$ denote the orthogonal projection onto the subspace $S_1 \subset \mathbb{R}^n$, and correspondingly, we take $P_2$ to be the orthogonal projection onto the subspace $S_2 \subset \mathbb{R}^n$....
gb628782's user avatar
  • 123
2 votes
2 answers
253 views

Proof (or reference) that the measure of a set $S\subset\mathbb{R}^n$ is preserved by an orthogonal transformation $V$?

Suppose $S\subset \mathbb{R}^n$ is measurable and has $\text{Vol}(S)$. If $M$ is a linear transformation then it is known that $\text{Vol}(MS)$ is equal to $\text{Vol}(S)$ multiplied by the product of ...
Jagerber48's user avatar
  • 1,701
0 votes
1 answer
69 views

Can $e^A$ be visualized as a rotation in $\mathbb{R}^3$ when $A$ is a $3 \times 3$ real skew-symmetric matrix?

I know that for any skew-symmetric matrix $A$, the exponential $e^A$ is an orthogonal matrix due to the identity: $(e^A)^T = e^{-A} = (e^A)^{-1}$ However, I’m looking for a deeper geometric intuition: ...
F. A. Mala's user avatar
  • 3,615
1 vote
0 answers
58 views

Find orthogonal $O$ such that $OMO'$ has constant diagonal

Let $M$ be a given symmetric real matrix. Can we always find an orthogonal matrix $O$ such that $$OMO^\top$$ has constant diagonal? That is, $(OMO^\top)_{ii}=d$ for some constant $d$ independent of $I$...
a06e's user avatar
  • 7,129
0 votes
0 answers
22 views

$SO(d)$ invariant functions are $O(d)$ invariant for narrow matrix multiplication

Background Let $ d, n \geq 2$ be natural numbers. Consider the groups $$ O(d)=\left\{R \in \mathbb{R}^{d \times d} \mid \quad R R^{T}=I_{d}\right\}, \quad S O(d)=\{R \in O(d) \text { and } \...
Michael's user avatar
  • 421
0 votes
0 answers
58 views

How to make sure that an orthogonal transformation results in only non-negative off-diagonal elements?

This is a very specific question but I have no idea on where to start. Let's say I have an orthogonal transformation $A = VB V^T$, where $B$ is a diagonal matrix and $V$ is an orthogonal matrix. For a ...
Stephphen's user avatar
2 votes
3 answers
124 views

number of integer vector near a hyperplane.

Let $x \in \mathbb{Z}^N$ be a non-zero integer vector. I am interested in obtaining an upper bound for the following quantity $$ \# \left\{ y \in \mathbb{Z}^N: \|y\| < Y, \operatorname{dist} \left(...
Johnny T.'s user avatar
  • 3,035
1 vote
1 answer
101 views

The order 4 isometry of the tetrahedron

Consider a tetrahedron centered at the origin. For example with the vertices $ (\pm1,0,-1/\sqrt{2}),(0,\pm1,1/\sqrt{2})$. The isometry group of the tetrahedron is $ S_4 $ and the subgroup of ...
Ian Gershon Teixeira's user avatar
0 votes
0 answers
49 views

Why does the orthogonal matrix act from the right in the Procrustes problem?

I have a problem with understanding the Procrustes optimization task, that is finding matrix $Q$ which transforms vectors stored in matrix $B$ so that the transformed vectors $BQ$ are as close as ...
hamsa's user avatar
  • 1
6 votes
2 answers
314 views

What is the smallest continuous group that contains $S(N)$?

We know that $O(N)$, the group of orthogonal matrices of dimension $N$, contains $S(N)$ as a subgroup, the group of permutations of $N$ elements. Another example is $U(N)$, the group of unitary ...
a06e's user avatar
  • 7,129
0 votes
0 answers
69 views

Which double covers of Lorentzian orthogonal groups restrict to spin groups?

It is known there are in general $8$ irreducible double covers of any Lorentzian orthogonal group $\operatorname{O}(1,q) \cong \operatorname{O}(q,1)$ for $q > 2$. These $8$ covers correspond to the ...
Craig's user avatar
  • 1,016
0 votes
1 answer
99 views

Eigenvalues of a complex orthogonal matrix

Property 1:A complex orthogonal matrix must have eigenvalues with modulus 1. Property 2: If all entries in the matrix are real (real orthogonal matrix), then the eigenvalues must be $\pm 1$ Proof of ...
Starlight's user avatar
  • 2,674
1 vote
0 answers
130 views

How to solve $\dot{A}(t)=A(t)B(t)$

I have a matrix differential equation $\dot{A}(t) = A(t)B(t)$ where $A$ is an orthonormal matrix and $B$ is a known skew-symmetric matrix, both 3-by-3. The farthest I could get in finding a general ...
Sebastian Mostek's user avatar
1 vote
1 answer
97 views

Properties of a matrix such that $A+A^T= \alpha Id$ [closed]

What can be said about the properties of an $n\times n$ invertible matrix $A$ with real entries such that $A+A^T=\alpha I_n$, where $\alpha\in \mathbb{R}$? Particularly, I am looking for properties ...
user392559's user avatar
  • 1,069

15 30 50 per page
1
2 3 4 5
79