69
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 ...
- 156k
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 $k$.
- 18.5k
28
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 ...
- 108k
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 $...
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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 $N$ ...
- 2,753
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
$$
S^{n-k-1}\to V(k+1,n)...
- 35.9k
16
votes
Accepted
Measuring the "distance" of a matrix from a diagonal matrix
Q: Is there a measure that captures the overall "level of orthogonality" of two matrices $A$ and $B$.
You can collect the $N^2$ elements of $A$ and $B$ into a pair of vectors $a$, $b$, and ...
- 188k
15
votes
Accepted
Is the linear span of special orthogonal matrices equal to the whole space of $N\times N$ matrices?
Elementary proof. The linear space $E$ spanned by $SO_n$ is the orthogonal of those matrices $M$ such that $\langle M,Q\rangle:={\rm Tr}(MQ)=0$ for every $Q\in SO_n$. Let $M=SR$ be a polar ...
- 52.3k
14
votes
A question about special linear group
These are exactly the finite-order elements:
If $G \in \text{SL}(n,\mathbb{Z})$ has finite order, then there exists an inner product preserved by $G$ (take an arbitrary inner product and add up its ...
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14
votes
$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns
There is no such matrix if $n\equiv 3\pmod 4$.
Suppose otherwise. Each column represents a vector of length $\sqrt n$. Since those vectors are pairwise orthogonal, their sum is a vector whose scalar ...
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13
votes
Accepted
Principal curvatures of $\mathbb{R}^{n^2}$-embedded SO(n)
What you are asking about is the second fundamental form of the embedding. Since this is a Lie group, it's enough to know what the second fundamental form is at the identity matrix $I_n=e$. Since ...
- 108k
13
votes
Accepted
Real orthogonal and sign
This is false already for $n=3$. A counterexample is given by the matrix
$$A=\frac{1}{3}\begin{bmatrix}
2 & 2 & -1\\
2 & -1 & 2\\
-1& 2 & 2
\end{bmatrix}.
$$
How did I ...
- 14.5k
13
votes
Accepted
Simple conjecture about rational orthogonal matrices and lattices
Proof
Let $R$ be any matrix. We have the obvious exact sequence
$$ 0 \longrightarrow\mathbb{R}^N \xrightarrow[\left(\begin{matrix} I \\ R \end{matrix}\right)]{} \mathbb{R}^N \oplus \mathbb{R}^N \...
12
votes
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 ...
- 2,139
12
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 ...
- 188k
12
votes
Accepted
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
The Skolem–Noether theorem says that $f$ arises by conjugation by an element of $\operatorname{GL}_n(\mathbb C)$ if and only if it also preserves matrix multiplication.
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12
votes
I want a smooth orthogonalization process
This is not possible as you formulate it. Consider the following two bases in $\mathbb R^3$:
\begin{gather*}
(e_1,\ e_1+\epsilon\, e_2) \\
(e_1,\ e_1+\epsilon\, e_3).
\end{gather*}
These are close ...
- 148k
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 ...
- 26.3k
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 (...
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11
votes
I want a smooth orthogonalization process
You want a function $f$ that assigns to each linearly independent $k$-tuple of vectors $B$ an orthonormal $k$-tuple of vectors $f(B)$ such that these span the same subspace. As others have pointed out,...
- 55.9k
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
$$3-{\rm Tr}(A^tB^tAB)\le2\omega^2(3-{\rm Tr}\,A)...
- 52.3k
9
votes
$2n \times 2n$ matrices with entries in $\{1, 0, -1\}$ with exactly $n$ zeroes in each row and each column with orthogonal rows and orthogonal columns
Your first conjecture was proven by Nate in the comments.
Your second conjecture is also true - there is no such matrix for $n=3$. If we just look at which entries are nonzero in each row, because any ...
- 148k
9
votes
Measuring the "distance" of a matrix from a diagonal matrix
A good tool for real symmetric positive-definite matrices is to let $d(A,B)=$ the square root of the sum of the squares of the natural logarithms of the generalized eigenvalues of $A$ and $B$; that is,...
- 203
9
votes
When is a linear isomorphism of $M_n(\mathbb{C})$ given by unitary conjugation?
As an alternative to LSpice's great answer:
Every linear map $f$ acting on $M_n(\mathbb{C})$ has a Choi matrix defined by
$$
C_f := \sum_{i,j}E_{i,j}\otimes f(E_{i,j}) \in M_n(\mathbb{C}) \otimes M_n(\...
- 6,351
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 ...
- 22.5k
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^...
- 156k
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 ...
- 55.9k
8
votes
I want a smooth orthogonalization process
The Gram–Schmidt process is locally smooth, i.e. there is a small neighborhood around every basis where you don't encounter the problem you mention and where the orthonromal basis from produced ...
- 8,597
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