All Questions
Tagged with orthogonal-matrices matrix-analysis
10 questions
1
vote
1
answer
151
views
How to prove that each element of $A(A^TA)^{-1}A^Ty$ is greater than 0, if $A(i,j)>0$ and $y=[1, 1, 1, ..., 1]^T$
Let $A\in \mathbb{R}^{m\times n}$, $m>n$, $rank(A)=n$, and $\forall 1 \leq i \leq m, 1 \leq j \leq n, A(i, j)>0$, $y=[1, 1, 1, ..., 1]^T$. Let $\beta=A(A^TA)^{-1}A^Ty$, how to prove that each ...
- 27
4
votes
0
answers
1k
views
Can an orthogonal matrix move monotonically toward a signed permutation matrix?
The question is motivated by this question on Mathematics SE.
Let $A \in O(n)$ be an orthogonal matrix that is not a signed permutation matrix, and let $P$ be the nearest signed permutation matrix to $...
- 209
0
votes
0
answers
33
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 ...
- 461
6
votes
1
answer
370
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 ...
- 6,741
11
votes
1
answer
565
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{...
- 6,741
16
votes
7
answers
2k
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 ...
- 355
3
votes
0
answers
298
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 ...
- 323
6
votes
2
answers
236
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 $$O_A=A(\...
- 6,741
8
votes
2
answers
4k
views
The space of positive definite orthogonal matrices
The matrix $\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}$ is orthogonal and indefinite.
$\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix}$ is positive definite and not orthonormal.
and the ...
- 720
19
votes
4
answers
825
views
How much redundancy resides in an $n \times n$ orthogonal matrix?
Suppose one has an $n \times n$ orthogonal matrix $M$:
$$
\left(
\begin{array}{ccc}
0.239326 & 0.846726 &
0.475161 \\
0.768893 & 0.13356 &
-0.625272 \\
0.592897 & -0....
- 151k