Nonzero element positions of a matrix

Update: The property "AdjacencyLists" gives what you need:

SparseArray[m]["AdjacencyLists"]

{{1, 2}, {1, 3}}

This approach, unlike the one using GatherBy in the original post, gives the empty set {} for the rows consisting entirely of zeros:

SparseArray[{{1, 2, 0}, {0, 0, 0}, {4, 0, 9}}]["AdjacencyLists"]

{{1, 2}, {}, {1, 3}}

Original post:

m = {{1, 2, 0}, {4, 0, 9}};
sa = SparseArray[m]["NonzeroPositions"];
GatherBy[sa, First][[All, All, -1]]

{{1, 2}, {1, 3}}

If the matrix isn't that sparse, then Pick may be a more efficient idea:

Pick[Range@Length@m[[1]], #, 0] & /@ UnitStep@-m

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