Efficient approximation of a matrix and its inverse

Assume that $A$ is a real $n\times n$ matrix whose rows constitute an orthonormal basis of $\mathbb R^n$.

Informal statement of question: Assume we want to approximate $A$ by a rational matrix, such that each entry can be written efficiently (that is, has a small binary encoding), but we require also the inverse of the approximate matrix to have small representation. Is this possible?

Formal statement of question: Let $p(n)$ be some polynomial in $n$. For a real number $r$, we say that $a/b$ is a polynomial approximation of $r$, if $a/b$ is a rational number (that is, $a,b$ are integers) and both $a$ and $b$ are of size at most $p(n)$ (e.g., their binary representation is of logarithmic size in $n$), such that $|r-a/b|\le 1/p(n)$.

Question: Does there exist a rational matrix $B$, such that $B$ polynomially approximates $A$ (that is, the entry $B_{ij}$ in $B$, is a polynomial approximation of the entry $A_{ij}$ in $A$, for all $1\le i,j\le n$), and such that $B^{-1}$ is a rational matrix whose entries are all polynomially-bounded (that is, for any $1\le i,j\le n$, $B^{-1}_{ij}=a/b$, where $a,b$ are integers of size at most $p(n)$) ?

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In $\mathbb{R}^3$, Milenkovic and Milenkovic give an alogrithm for efficiently approximating an orthogonal matrix by a rational orthogonal matrix. As lhf points out, the inverse of an orthogonal matrix is its transpose, so the inverse will also have short entries in this setting.

Regarding $n>3$, here is a tentative thought, and a reference. I haven't put much effort into either :).

Let $v=(v_1, v_2, \ldots, v_n)$ be a nonzero vector. Define a linear operator $$s_v(u) := u - 2 \frac{\langle v,u \rangle}{\langle v,v \rangle} v.$$ This is the orthogonal reflection that negates $v$. Note that, if $v \in \mathbb{Q}^n$, then the entries of the matrix $s_v$ are rational. This is true even if $v$ does not have norm $1$.

Now, any rotation matrix can be written as a product of $\leq n$ reflections: $R=\prod_{i=1}^h s_{v_i}$ for some sequence of vectors $v_i$ in $\mathbb{R}^n$. A potential algorithm, then, is to find such a factorization and then approximate each $v_i$ by a rational vector $w_i$ which is roughly parallel to it. (There are plenty of standard algorithms for rational approximation of a vector.) Then take $\prod s_{w_i}$ as the approximation to $R$.

I got this strategy from a paper of Eric Schmutz. Schmutz follows this strategy, but he forces his approximating vectors $w_i$ to lie on the unit sphere. As far as I can see, this is a waste of effort, since $s_v$ is orthogonal with rational entries even if $v$ is not on the unit sphere. However, Schmutz has exact bounds, which you may find useful.

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Thanks, I'll have a look. Maybe they (or future citations of this paper) have some references for the general case of $n$. – Iddo Tzameret Jun 30 '10 at 18:09
I haven't followed through on the links, so I don't know which method do they use, but my first reaction to "rational orthogonal matrix" is "Cayley transform". – Victor Protsak Jul 1 '10 at 5:45

If A is orthogonal then its inverse is the transpose and so you only need to approximate A.

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But your approximation may not be orthogonal, so its inverse may require a lot of bits to store. – David Speyer Jun 30 '10 at 17:45
Yes, David is right. (lhf would be right too, if the orthogonalization algorithm (of e.g., Gram-Schmidt) would end up with a matrix in which the entries are polynomially-bounded by the entries in the original matrix. I can't see why this should be true though.) – Iddo Tzameret Jun 30 '10 at 18:01