DenseMatrix

DenseMatrix is a two-dimensional array-like data structure that stores every element in a rectangular grid, typically numerical values. It represents a mathematical matrix where no entries are elided; it is contrasted with sparse representations that store only nonzero entries.

Storage and layout: In many programming environments, a DenseMatrix is backed by a contiguous array of numbers, with either row-major (row by row) or column-major (column by column) order. The element at row i, column j is stored at a predictable offset. Data types usually include real or complex floating-point numbers.

Operations and algorithms: DenseMatrix supports standard operations such as addition, subtraction, scalar multiplication, matrix multiplication, transposition, and decompositions (LU, QR, Cholesky), as well as retrieval of determinants, inverses, and eigenvalues in appropriate cases. Computing with a DenseMatrix often uses optimized numerical libraries (for example, BLAS) to exploit vectorization and cache locality.

Complexity and performance: Memory consumption is O(nm) for an n-by-m matrix, which can be substantial for large sizes. Time complexity for multiplication is O(n^3) in the naive case, with improved algorithms in optimized libraries.

Use and alternatives: DenseMatrix is common in numerical computation, data analysis, and machine learning. It is contrasted with sparse matrices, which store only nonzero entries to save memory when the matrix is sparse. Some ecosystems provide a dedicated DenseMatrix type, while others implement dense data as a standard two-dimensional array or similar structure.

See also: Matrix (linear algebra), Sparse matrix.