  We define a set of operations called crystal operations on matrices with
entries either in {0,1} or in N. There are horizontal and vertical crystal
operations, giving rise to two commuting structures of a crystal graph on these
matrices. They provide a new perspective on many aspects of the RSK
correspondence and its dual, and related constructions. Under a straightforward
encoding of semistandard tableaux by matrices, the operations correspond to
crystal operations on tableaux, respectively to individual moves occurring
during a jeu de taquin slide. We show that the (dual) RSK correspondence and
the Burge correspondence arise as decompositions: a matrix M can be transformed
by crystal operations into each of the matrices encoding P and Q symbol
associated to M under these correspondences, and it can be reconstructed from P
and Q.
  These decomposition can also be interpreted as computing Robinson's
correspondence, or as the Robinson-Schensted correspondence for pictures. From
a particular way of applying crystal operations, the computation of these
decompositions by growth diagrams can be deduced, as well as the local rules
that are to be used. We show that that crystal operations leave a version of
Greene's poset invariant defined for matrices unchanged, so that for such
questions in the setting of matrices they can take the place of elementary
Knuth transformations on words.
