matrixcomponenten

Matrixcomponenten, in mathematics and linear algebra, refer to the individual entries of a matrix. An m×n matrix A is defined by its components a_ij, where i = 1,...,m denotes the row and j = 1,...,n denotes the column. Thus A = [a_ij]. Each a_ij is a scalar and together they determine all properties of A.

In addition to being single numbers, the rows and columns of a matrix are themselves components of higher‑level objects. Each row i forms a row vector (a_i1, a_i2, ..., a_in) and each column j forms a column vector (a_1j, a_2j, ..., a_mj). In linear transformations, the j-th column can be interpreted as the image of the j-th standard basis vector.

Matrix operations act componentwise: addition and subtraction add corresponding entries, and scalar multiplication multiplies each component by the scalar. Matrix multiplication combines rows and columns: the entry in row i, column j of AB equals the dot product of row i of A with column j of B.

For square matrices, diagonal components a_ii play a special role; the trace is the sum of diagonal components. Other global properties, such as determinant and rank, are not single components but properties of the entire arrangement of components.

In practice, components are stored in memory as a two-dimensional array; many computations use vectorization by flattening the matrix into a one-dimensional vector. Components thus serve as the fundamental building blocks for representing, manipulating, and analyzing matrices in both theory and applications.