Existence of Rational Orthogonal Matrices

Question:

Let $A\in\mathbb{R}^{n \times n}$ be an orthogonal matrix and let $\varepsilon>0$. Then does there exist a rational orthogonal matrix $B\in\mathbb{R}^{n \times n}$ such that $\|A-B\|<\varepsilon$?

Definitions:

• A matrix $A\in\mathbb{R}^{n \times n}$ is an orthogonal matrix if $A^T=A^{-1}$
• A matrix $A\in\mathbb{R}^{n \times n}$ is a rational matrix if every entry of it is rational.
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Yes. It is a theorem of Cayley that the mapping $S \rightarrow (S-I)^{-1}(S+1)$ gives a correspondence between the set of $n\times n$ skew-symmetric matrices over $\mathbb{Q}$ and the set of $n\times n$ orthogonal matrices which do not have one as an eigenvalue. Since the mapping is nice, and rational skew-symmetric matrices are dense in the set of all skew-symmetric matrices, you have your result. For more, see the very nice paper by Liebeck and Osborne

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That was my answer as well. I would also recommend Weyl's gem, The Classical Groups. Their Invariants and Representations. He has a whole section devoted to the Cayley parameterization. (Chap. II. Sec II) –  Liviu Nicolaescu Mar 2 '12 at 20:55

Sure. Consider matrices which fix $n-2$ of the standard basis vectors and describe a rotation in the plane spanned by the last two about an angle $\theta$ such that $\sin \theta, \cos \theta$ are both rational; these are dense in all such rotations, and all such rotations generate the orthogonal group, so the corresponding products (all of which are rational) are dense in the orthogonal group.

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By the way, the elementary generators that you describe are known as "Givens rotations". –  Federico Poloni Mar 2 '12 at 20:47

I should say yes. For this, I shall use the fact that in the unit sphere $\mathbb S^{d-1}$, the set of rational vectors is dense. I shall proceed by induction over $n$.

So let $A\in {\bf O}_n(\mathbb R)$ be given. Let $\vec v_1$ be its first column, an element of ${\mathbb S}^{n-1}$. We can choose a rational unit vector $\vec w_1$ arbitrarily close to $\vec v_1$. The first step is to construct a rational orthogonal matrix $B$ with first column $\vec w_1$. To this end we choose inductively rational unit vectors $\vec w_2,\ldots,\vec w_n$. This is possible because at each step, we may take a rational unit vector in the unit sphere of a "rational" subspace. Here, a subspace $F$ is rational if it admits a rational basis.

Now, let us form $A_1=B^{-1}A$. This is a orthogonal matrix, whose first column is arbitrarily close to $\vec e_1$. Hence its first line is close to $(1,0,\ldots,0)$ as well. Thus $$A_1\sim\begin{pmatrix} 1 & 0^T \\\\ 0 & R \end{pmatrix}.$$ The matrix $R$ is arbitrarily close to ${\bf O}_{n-1}({\mathbb R})$. By the induction hypothesis, there exists a rational orthogonal matrix $Q$ arbitraly close to $R$. Then $$B\begin{pmatrix} 1 & 0^T \\\\ 0 & Q \end{pmatrix}$$ is arbitrarily close to $A$.

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