Rational Canonical Form for Fields of positive characteristic Recently, I was working on the rational canonical forms of matrices and then one query popped up which i didn't manage to find in the literature (that I am aware of course). So, if $k$ is an infinite field of characteristic zero then every matrix with entries over $k$ has a rational canonical form. Do you know what happens when the base field  is of positive characteristic? Do we have any variations or restrictions in that case? Does there exist any discrepancy if $k$ is algebraically closed (which means that is infinite)?

To the best of my knowledge, the characteristic of k plays no role at all.  I.e. if k is any field, then k[X] is a Euclidean domain, and if A is an n by n matrix over k, then the characteristic matrix X.I-A defines a presentation of the k[X] module structure defined by A on k^n.  Hence diagonalizing the matrix X.I-A by row and column operations, yields a matrix whose monic diagonal entries give a product decomposition for k^n into cyclic A-invariant subspaces.  On each of these cyclic factor spaces A has as matrix, for any choice of cyclic vector, a "companion" matrix.  Such a decomposition is what I understand as a rational canonical form.  One can even diagonalize X.I-A so the diagonal entries successively divide each other, and this gives more properly the (unique) rational canonical form.  A superb reference is Van der Waerden, Modern Algebra, vol. 2.
