tätmatris

Tätmatris is Swedish for “dense matrix.” In mathematics, a dense matrix is a matrix in which most of the entries are nonzero. It is typically contrasted with a sparse matrix, where a large portion of the elements are zero.

Definition and density: For an m by n matrix, density is the ratio of nonzero elements to the total number of entries (m·n). A matrix is treated as dense when this ratio is high, often near one, meaning that almost every position carries a meaningful value. In practice, the distinction influences storage and algorithm choices.

Storage and data layout: Dense matrices are usually stored as a two-dimensional array in memory, using either row-major or column-major order. The memory requirement is O(mn) elements. Because zeros are stored alongside nonzeros, dense storage is straightforward and benefits from contiguous memory layouts, which improve cache performance.

Operations and performance: Common operations on dense matrices include addition, subtraction, and multiplication. The time complexity for matrix addition is O(mn), while multiplication of an m×n matrix by an n×p matrix is O(mnp). Dense storage enables highly optimized numerical routines (such as BLAS and LAPACK) that leverage cache locality and vectorized instructions.

Applications and examples: Dense matrices arise in many areas, including solving linear systems, eigenvalue problems, numerical simulations, computer graphics, and neural networks with dense weight matrices. A fully connected graph is represented by a dense adjacency matrix when every pair of nodes is connected.

Relation to sparse matrices: For data with many zeros, sparse representations reduce memory and compute. When density is high, dense methods are typically preferred due to simplicity and performance.