68
votes

Accepted

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

$\def\RR{\mathbb{R}}$ This problem was solved by
Whitehead, G. W., Note on cross-sections in Stiefel manifolds, Comment. Math. Helv. 37, 239-240 (1963). ZBL0118.18702.
Such sections exist only in ...

46
votes

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

Unless I'm missing something, I think that the hairy ball theorem states precisely that you cannot do this when $n = 3$ and $k = 1$. I'm not sure what happens for other values of $n$ and $k$.

27
votes

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

For $n>2$, the span of the matrices in $\mathrm{SO}(n)$ is the full space $M_n(\mathbb{R})$ of $n$-by-$n$ matrices with real entries.
One proof is using representation theory: If we let $S\subset ...

22
votes

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

The space of orthonormal $k$-frames in $\mathbb{R}^n$ is the Stiefel manifold $V(k, n) = SO(n)/SO(n - k)$. There is a natural $SO(k)$ action on $V(k, n)$ and the quotient is the oriented grassmannian $...

18
votes

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

Let $S$ be the span of $SO(N)$ .
Then it's obvious that if $A \in S$ and $D_1, D_2 \in SO(N)$ then $D_1^{-1} A D_2 \in S$ .
Therefore it's enough to show that $B := diag(1,0,...0) \in S$ .
If $N$ ...

17
votes

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

Denoting the Stiefel manifold of orthonormal $k$-frames in $\mathbb{R}^n$ by $V(k,n)$ as in Michael Albanese's answer, what you are asking for is a section of the sphere bundle
$$
S^{n-k-1}\to V(k+1,n)...

16
votes

Accepted

### Measuring the "distance" of a matrix from a diagonal matrix

Q: Is there a measure that captures the overall "level of orthogonality" of two matrices $A$ and $B$.
You can collect the $N^2$ elements of $A$ and $B$ into a pair of vectors $a$, $b$, and ...

14
votes

Accepted

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

Elementary proof. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar ...

14
votes

### A question about special linear group

These are exactly the finite-order elements:
If $G \in \text{SL}(n,\mathbb{Z})$ has finite order, then there exists an inner product preserved by $G$ (take an arbitrary inner product and add up its ...

14
votes

### $2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns

There is no such matrix if $n\equiv 3\pmod 4$.
Suppose otherwise. Each column represents a vector of length $\sqrt n$. Since those vectors are pairwise orthogonal, their sum is a vector whose scalar ...

13
votes

Accepted

### Real orthogonal and sign

This is false already for $n=3$. A counterexample is given by the matrix
$$A=\frac{1}{3}\begin{bmatrix}
2 & 2 & -1\\
2 & -1 & 2\\
-1& 2 & 2
\end{bmatrix}.
$$
How did I ...

13
votes

Accepted

### Simple conjecture about rational orthogonal matrices and lattices

Proof
Let $R$ be any matrix. We have the obvious exact sequence
$$ 0 \longrightarrow\mathbb{R}^N \xrightarrow[\left(\begin{matrix} I \\ R \end{matrix}\right)]{} \mathbb{R}^N \oplus \mathbb{R}^N \...

12
votes

Accepted

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

What you are asking about is the second fundamental form of the embedding. Since this is a Lie group, it's enough to know what the second fundamental form is at the identity matrix $I_n=e$. Since ...

12
votes

### Real orthogonal and sign

Here's a simple argument that this is false for $n > 2$.
In dimension $n$ there are $2^{n}$ orthants, $2^{n-1}$ if one considers them modulo sign. A pair of antipodal orthants means a pair of ...

11
votes

Accepted

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

Orthonormal $\boldsymbol n$-frames : https://en.wikipedia.org/wiki/Stiefel_manifold.
Added: This terminology of Hirzebruch (1966), Steenrod (1951) translates the $\boldsymbol n$-Systeme of Stiefel (...

11
votes

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

Clearly the span of $SO(n)$ contains the tangent space of $SO(n)$, the antisymmetric matrices. But it also contains the rotations by $\pi$ in any 2-plane, so contains the diagonal matrices with even ...

10
votes

### Measuring the "distance" of a matrix from a diagonal matrix

I read the question differently, so at least one of us must have misunderstood your question. 😅
Q: Is there a measure that captures how close a matrix $A \in \mathbb{R}^{n \times n}$ is to being ...

9
votes

Accepted

### matrix inequality with orthogonal matrices

I confirm Peter's suggestion that the best constant is $\frac1{\sqrt2}$. For let $\omega$ be this best constant. Then the inequality amounts to writing
$$3-{\rm Tr}(A^tB^tAB)\le2\omega^2(3-{\rm Tr}\,A)...

9
votes

### $2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns

Your first conjecture was proven by Nate in the comments.
Your second conjecture is also true - there is no such matrix for $n=3$. If we just look at which entries are nonzero in each row, because any ...

9
votes

### Measuring the "distance" of a matrix from a diagonal matrix

A good tool for real symmetric positive-definite matrices is to let $d(A,B)=$ the square root of the sum of the squares of the natural logarithms of the generalized eigenvalues of $A$ and $B$; that is,...

8
votes

### matrix inequality with orthogonal matrices

Numerical experiments suggest that actually $\|AB-BA\|_F\leq\frac{1}{\sqrt{2}} \|A-I\|_F\|B-I\|_F$ could be true, and that this bound is sharp.
I don't know if that helps, but this sharpened ...

8
votes

### $2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns

$\def\Id{\text{Id}}$A Hadamard matrix is an $n \times n$ matrix with entries in $\{-1, 1 \}$ with $H H^T = H^T H = n \Id$. What you want is a matrix $X$ with entries in $\{-1, 0, 1 \}$ with $XX^T = X^...

8
votes

Accepted

### Orthogonal basis of ${\bf Sym}_n(\mathbb R)$, made of orthogonal matrices

Here is an example with $n=4$. (ADDED BELOW: An example for any power of $2$)
I identify $\mathbb R^4$ with the quaternions, and describe $10$ subspaces such that any two of the resulting orthogonal ...

8
votes

### Computing Haar measure of matrices sampled from SO(n)

Indeed, the distribution function of the eigenphases of a random matrix in $\operatorname{SO}(n)$ has a peak at 0 and at $\pm\pi$. It only becomes uniform for large $n$. The joint distribution ...

7
votes

Accepted

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

The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational ...

7
votes

### Existence of parametrizations of rational orthogonal matrices

The formula in @Carlo Beenakker answer does not give all rational orthogonal matrices, but only those for which $-1$ is not an eigenvalue. Here is a rational parametrization which gives all orthogonal ...

7
votes

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

I don't know how much you want, but the moment you write things in terms of multilinear algebra, everything seems to become pretty transparent.
In general, let $V$ be an $n$-dimensional real vector ...

7
votes

Accepted

### A subgroup of $\mathrm{SL}_n(\mathbb{Z}/p\mathbb{Z})$

This is the special orthogonal group over the field $\mathbb{Z}/p \mathbb{Z}$. Or, rather, one of the "special orthogonal groups". For any invertible symmetric matrix $Q$, one can consider ...

7
votes

### Is it true $\left\|\log(RS)\right\|≤\left\|\log(R)+\log(S)\right\|$ for all $R,S \in \mathrm{SO}(3)$, where $\|\cdot\|$ is the Frobenius norm?

I realized a problem with my 'counterexample', so I no longer claim that the desired inequality does not hold on $\mathrm{SO}(3)$. The proof that it does hold on the quaternions is still OK. I'll ...

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