We explicitly compute the celebrated Robinson--Schensted--Knuth
bijection (RSK) between the set of the matrices with non-negative
integer entries, and the set of the plane partitions. More
precisely, in suitable linear coordinates on both sets, the RSK
is expressed via minima of linear forms, i.e, in piece--wise
linear terms. In particular, we answer the following question by
C.~Greene and G.~Viennot: ``What shape corresponds to a given
matrix under the Robinson--Schensted--Knuth correspondence?" Our
main tools in establishing these formulae are the quantum
matrices and crystal bases. As a byproduct of our approach, we
compute the corresponding crystal equivalence in terms of
"generalized" Kazhdan--Lusztig polynomials.
