Construct a sparse matrix with values at only a few specified positions:

View it as a matrix:

Convert it to an ordinary dense matrix:

Make a large sparse vector:

Make a large sparse matrix:

Make a large sparse depth-3 array:

Construct a tridiagonal matrix using patterns for indices:

Construct a 10,000×10,000 version:

Make a sparse diagonal matrix:

This is equivalent to DiagonalMatrix:

Except that as a sparse matrix, it uses much less memory:

Construct a block diagonal matrix using rules with Band:

Convert an ordinary matrix into a sparse matrix:

Make a rank-4 sparse tensor with values at random positions:

ArrayRules produces the minimal list of rules needed to specify the SparseArray:

Many typical operations work with SparseArray objects as they would for equivalent lists:

Arithmetic works elementwise just as it does for lists:

Matrix products are done with Dot:

Many linear algebra functions are done efficiently with the sparse form:

Many other list commands work automatically:

The unspecified elements can have any value:

Construct a sparse matrix with all machine-number values:

Construct a sparse matrix with exact integer values:

N[s] is the same as ns:

Create a list with a single nonzero element:

Plot a list of rules:

Represent a network with an adjacency matrix:

Solve a boundary-value problem using finite differences:

A SparseArray object is Equal to the corresponding ordinary list:

They are not SameQ because the expression structure is different:

For functions f that work with SparseArray objects, typically f[s]==f[Normal[s]]:

This includes all functions with the attribute Listable:

Convert linear expressions to SparseArray objects using CoefficientArrays:

Convert from SparseArray to expressions using Dot:

If a position is repeated in the rule list for SparseArray, the first instance is used:

SparseArray objects can represent data too large to represent in normal form:

Using Normal will give a SystemException:

Sparse operations do not by default check for cancellation:

Use SparseArray to recompute the sparse structure:

The internal structure of a SparseArray representation is not unique and SameQ detects this:

Use SparseArray to recompute the sparse structure:

Note that Equal works as expected:

The internal structure of SparseArray representation is not unique and setting parts can change that structure:

Test if the SparseArray instances are the same:

Use SparseArray to recompute the sparse structure:

Note that Equal works as expected:

Operations with side effects may give different values when iterating over SparseArray:

With Reap and Sow you can see what elements are accessed:

For a SparseArray object, Part gives parts of the represented list:

The FullForm is a way of reconstructing the object from basic storage information:

A SparseArray object is treated as atomic for functions that do not work on the representation:

Cases does not work on the represented matrix:

You can often use the result of ArrayRules to get the information without expanding:

Even for a dense SparseArray there can be a division by 0:

The density of a SparseArray is 1:

Since a density 1. SparseArray is a very special case, making this work would be computationally excessively expansive. In this case, the use of Normal is justified as the matrix is dense:

The Game of Life: