In functional programming, sparse matrices can be effectively represented using various data structures, such as lists of lists, maps (or dictionaries), and tuples.

A sparse matrix is defined as a matrix in which the majority of its elements are zeros. In a functional programming language, there are several ways to represent a sparse matrix. One common approach is to use a list of lists, where each inner list corresponds to a row of the matrix. However, this representation can be inefficient for very large matrices that contain numerous zeros, as it consumes memory to store these zero values.

A more memory-efficient method for representing sparse matrices is to utilize a map or dictionary. In this approach, you only store the non-zero elements along with their respective positions. For instance, you could implement a map where the keys are tuples representing the positions of the non-zero elements, while the values correspond to the non-zero elements themselves. This strategy significantly reduces memory usage, particularly for large matrices that predominantly consist of zeros.

Another alternative is to use a list of tuples, with each tuple containing the row index, column index, and value of a non-zero element. This method is similar to the map representation, but instead of a map, you maintain a list of tuples. This can prove to be more efficient than the map method, especially when dealing with very large matrices where non-zero elements are sparsely distributed, as it avoids the overhead of storing keys.

When working with sparse matrices in functional programming, you typically employ higher-order functions such as ‘map’, ‘filter’, and ‘reduce’ to manipulate the data. For example, you can use the ‘map’ function to apply a specific operation to all non-zero elements, the ‘filter’ function to extract non-zero elements that satisfy particular conditions, and the ‘reduce’ function to combine all non-zero elements in a specified manner.