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We consider random symmetric matrices with independent entries distributed according to the Haar measure onZ  p   $\mathbb {Z}_p$    for odd primes  p  and derive the distribution of their canonical form with respect to several equivalence relations. We give a few examples of applications including an alternative proof for the result of Bhargava, Cremona, Fisher, Jones and Keating on the probability that a random quadratic form overZ  p   $\mathbb {Z}_p$    has a non‐trivial zero.

On the distribution of equivalence classes of random symmetric p ‐adic matrices