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## Hot answers tagged orthogonal-matrices

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 ...
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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 $k$.
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27 votes

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### 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 $N$ ...
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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 ...
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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 ...
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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

### 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 ...
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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)...
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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 ...
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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,...
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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

$\def\Id{\text{Id}}$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 $XX^T = X^... • 152k 8 votes Accepted ### Orthogonal basis of${\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 ... • 54.5k 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$n. The joint distribution ... • 180k 7 votes Accepted ### What's the best orthonormal matrix to align two matrices in the operator norm sense? The operator norm version of this problem is considered in: The solution of orthogonal Procrustes problems for a family of orthogonally invariant norms, by G. A. Watson, Advances in Computational ... • 28.4k 7 votes ### Existence of parametrizations of rational orthogonal matrices The formula in @Carlo Beenakker answer does not give all rational orthogonal matrices, but only those for which-1$is not an eigenvalue. Here is a rational parametrization which gives all orthogonal ... • 89.1k 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 ... • 2,185 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 ... • 152k 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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