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Let $A$ and $B$ two symmetric matrices definite positive over $\mathbb{R}$. Then we say that $A$ and $B$ are integrally equivalent if there exists $Q\in GL_n(\mathbb{Z})$ such that

$A=Q.B.Q^t$ (1)

My question is: does it exist a characterization of such equivalence classes?

In other words, if I know matrices $A$ and $B$, how can I see if these matrices are related by equation (1)?

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I think you probably can decide this without tears, but just because your matrices are positive definite. Notice that, if $x$ is a row vector of integers, $$ x A x^T = x Q B Q^T x^T = y B y^T, $$ where we take $y = x Q.$ So, your two matrices have the same determinants, but they also have the same $\mathbb Z$-minimum. Indeed, I think they have the same "successive minima," and I would not be surprised if that were sufficient as well. In any case, you ought to get Gruber and Lekkerkerker. Note that emaphasis is switched from the matrix $A$ to the compact set with nonempty interior, $$ x A x^T \leq 1. $$

Now, why is is possible to find the $\mathbb Z$-minimum, which is some peculiar real number in your setting? Begin with $ m $ being the smallest diagonal entry of $A.$ The set of integer vectors $x$ with $ x A x^T \leq m$ is bounded by eigenvalue considerations (I actually do a little thing with Lagrange multipliers to get a separate bound for each coordinate $x_i.$) This gives a cube or rectangular box depending how you do it, leading to a finite set of integer vectors. So finding the $\mathbb Z$-minimum is a finite check.

I have seen this problem in infinite detail only for indefinite binary forms, see Cusick and Flahive

Alright, I am remembering things. Minkowski reduction is not definitive even for integral quadratic forms for dimensions at least three. It is quite good in dimensions 3,4. Alexander Schiemman made an improvement to Eisenstein's reduction in dimension 3, and wrote it up allowing for real entries, Mathematische Annalen, volume 308 (1997) pages 507-517, Ternary Positive Definite Quadratic Forms are determined by their Theta Series.

EDIT, 3:52 pm Pacific: Some of Schiemann's favorite themes give the simplest view of this. By definiteness, you can make two lists of represented values up to some bound, where "represented" means by an integral vector $x$ as above. Although these are real numbers, let me still call each list a "theta series," because we are able to count the number of times each real number is represented. Now, it is necessary that the two matrices have the same "theta series" in order to be $\mathbb Z$-equivalent. However, if the dimension is 4 or larger, this is not sufficient. The original examples of disagreement (same theta series but not equivalent) were by Witt, dimension 16, later used by Milnor. The general topic is loosely under the heading of "Can you hear the shape of a drum?" Anyway, there was progress lowering the dimension, especially by Kneser and then Kitaoka. Finally, Schiemann found a four-dimensional pair, Arch. Math. 54 (1990) pages 372-375. Conway was able to include a version in his book The Sensual Quadratic Form. A less Conwayish description, including just coefficients, in in Quaternary Quadratic Forms by Gordon L. Nipp. There we go, page 1 and then again on page 110, in genus 4: the final two forms with minimum 2, after which there are three forms with minimum 3 and that is the end of that genus and that discriminiminiminant.

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    $\begingroup$ Last I heard, deciding equivalence between two positive-definite quadratic forms of order $n$ is feasible for a given $n$, but the known algorithms involve constants that explode with $n$, and the problem is expected to be computationally intractable for large $n$. So is finding the minimal nonzero norm (and thus a fortiori all the successive minima). Thus no computationally feasible characterization is expected for integral equivalence, though there are some easily checked necessary conditions, and in practice one can (usually?) handle $n$ up to at least $24$ or so. $\endgroup$
    – Noam D. Elkies
    Commented Apr 19, 2012 at 5:26
  • $\begingroup$ Apart from the usual theta series which counts the vectors of given norm, there are also the degree $k, 1 \le k \le n$ theta series which counts the number of $k$ dimensional sublattices of given volume. I have always wondered if there are non-equivalent positive-definite positive definite integral quadratic forms with the same degree $k$ theta series for $k$ up to $n$ ? For example, does Schiemann's 4d example has the same Siegel 2,3,4 theta series ? $\endgroup$
    – Chua KS
    Commented Apr 19, 2012 at 7:11
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    $\begingroup$ Certainly two different 4D lattices cannot have the same Siegel-4 theta series! (Siegel $k$ counts sublattices of rank up to $k$ by isomorphism class, not just volume.) I'm told that the two unimodular lattice of rank $16$ have the same $\Theta_k$ not just for $k=1$ but also for $k=2$ and $k=3$, but not for $k=4$ where the difference is the modular form that vanishes on the Jacobian locus. $\endgroup$
    – Noam D. Elkies
    Commented Apr 19, 2012 at 13:17
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    $\begingroup$ I heard that the remarkable fact that the smallest $k$ for which Siegel gamma discernates between $E_8\perp E_8$ and $E_{16}$ is $k=4$ is due to Kneser. $\endgroup$
    – few_reps
    Commented Jun 24, 2014 at 19:58
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    $\begingroup$ @Jagy : well, it seems to be this : link.springer.com/article/10.1007%2FBF01361543 $\endgroup$
    – few_reps
    Commented Jun 24, 2014 at 20:45
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An answer to your question is given in G.F. Voronoi (1908). "Nouvelles applications des paramètres continus à la théorie de formes quadratiques". Journal für die reine und angewandte Mathematik 134: 198–287.

In this wonderful work, Voronoi explains how to identify a fundamental domain for the action of $\mathrm{GL}_n(\mathbf Z)$ on the space of positive definite quadratic forms. All I can say is hat it is worth reading it (a more contemporary reference is Martinet : Perfect latticess in Euclidean spaces, Grundlehren der mathematischen Wissenschaften (Book 327), 2003, which is also an excellent reference).

But as Noam Elkies wrote above, this method cannot be used in high dimensions (for now, it has been done in dim. <=8).

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