Get the explicit elements in a SparseArray:

These rules are sufficient to efficiently construct an identical SparseArray:

The last element of ArrayRules[s] is always {_,_,…}->def:

A SparseArray with a default value of 2:

You can override this by explicitly specifying what default you would like:

These will construct a SparseArray identical to SparseArray[m,Automatic,1]:

Positions of 1 in an explicit array with the default taken to be 0:

These will construct a SparseArray identical to SparseArray[a]:

Positions of 0 with 1 taken as default:

These will construct a SparseArray identical to SparseArray[a,Automatic,1]:

Get the number of explicit elements in a SparseArray:

Get the explicit elements of a sparse array satisfying a condition:

Note the more complicated pattern is needed since Cases has special behavior for Rule:

SparseArray objects with positive and negative values:

Get the upper and lower triangular parts of a sparse matrix:

Lower triangular part with 1s on the diagonal:

This just happens to be the LU decomposition of a tridiagonal matrix:

Make a plot showing the positions of the explicit elements of a SparseArray with tooltips:

MatrixPlot generally makes a visually better plot:

For a SparseArray s, SparseArray[ArrayRules[s],Dimensions[s]] is identical to s:

Specifying the dimensions is needed since they would be inferred from explicit elements:

For an explicit array ArrayRules can be written in terms of Position:

This will not work for SparseArray objects because pattern matching works on the FullForm: