  We present a novel recursive algorithm for reducing a symmetric matrix to a
triangular factorization which reveals the rank profile matrix. That is, the
algorithm computes a factorization $\mathbf{P}^T\mathbf{A}\mathbf{P} =
\mathbf{L}\mathbf{D}\mathbf{L}^T$ where $\mathbf{P}$ is a permutation matrix,
$\mathbf{L}$ is lower triangular with a unit diagonal and $\mathbf{D}$ is
symmetric block diagonal with $1{\times}1$ and $2{\times}2$ antidiagonal
blocks. The novel algorithm requires $O(n^2r^{\omega-2})$ arithmetic
operations. Furthermore, experimental results demonstrate that our algorithm
can even be slightly more than twice as fast as the state of the art
unsymmetric Gaussian elimination in most cases, that is it achieves
approximately the same computational speed. By adapting the pivoting strategy
developed in the unsymmetric case, we show how to recover the rank profile
matrix from the permutation matrix and the support of the block-diagonal
matrix. There is an obstruction in characteristic $2$ for revealing the rank
profile matrix which requires to relax the shape of the block diagonal by
allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient.
This relaxed decomposition can then be transformed into a standard
$\mathbf{P}\mathbf{L}\mathbf{D}\mathbf{L}^T\mathbf{P}^T$ decomposition at a
negligible cost.
