A sparse matrix is a matrix that has a lot of zeros in it. In other words, most of the elements in a sparse matrix are zeros, and only a small number of elements actually contain useful values. For example, imagine a 100x100 matrix. If only a few of the cells have values, and the rest are all zero, that's a sparse matrix. Sparse Matrix A sparse matrix is a type of matrix that contains mostly zero elements, with only a few non-zero elements. In a sparse matrix, the majority of the entries are zeros, which makes it different from a dense matrix, where most of the elements are non-zero. Characteristics of a Sparse Matrix: 1. High proportion of zeros: The defining characteristic of a sparse matrix is that most of its elements are zeros. 2. Efficient storage: Since a sparse matrix contains a large number of zero elements, storing all these zeros is wasteful. Therefore, special techniques are used to store only the non-zero elements, saving memory and computational resources. 3. Memory efficiency: Instead of storing every element, sparse matrices are typically stored using data structures that only keep track of the non-zero elements and their positions. This reduces memory usage significantly, especially for large matrices. Example 0 0 0 1 0 0 2 0 0 3 0 0 4 0 0 0 In this matrix, most of the elements are zeros, and only four elements are non-zero. Thus, it is a sparse matrix. Storage of Sparse Matrices: To store a sparse matrix efficiently, we use methods that only keep track of the non-zero elements. Here are a few common ways: 1. Coordinate List (COO) format: This method stores the non-zero elements as a list of coordinates. For each non-zero element, we store its row index, column index, and value. For the example above, the sparse matrix would be represented as: • (0, 3, 1) • (1, 2, 2) • (2, 1, 3) • (3, 0, 4) 2. Compressed Sparse Row (CSR) format: This format stores the matrix in three arrays: • An array of non-zero elements. • An array of column indices for each non-zero element. • A pointer array indicating the starting position of each row in the non-zero element array. 3. This method is particularly efficient for matrix operations like matrix-vector multiplication. Applications of Sparse Matrices: • Scientific computing: Sparse matrices are widely used in fields like physics and engineering, where large matrices are often generated, but only a small portion of the matrix contains meaningful data. • Machine learning: In machine learning, sparse matrices can be used for tasks like representing sparse datasets, such as text data where most words in a document are absent. • Graph representations: Sparse matrices are also used to represent graphs in adjacency matrices, where most entries are zeros, except for the connections between nodes.