Hermite Normal Form in “columns” convention

After describing the Hermite Normal Form (HNF), MathWorld explains:

The Hermite normal form for integer matrices is implemented in Mathematica as HermiteDecomposition[A], which however uses the “rows” convention (thus making H\mathbf H upper triangular) and replaces (the condition hij≤0h_{i\,j}\leq 0 and |hij|