Matritze

Matritze is a mathematical construct consisting of a rectangular array of elements arranged in rows and columns. Each element is typically denoted aij, with i indexing the row and j indexing the column. Matritzen (plural) are used to represent coefficients of linear systems and to encode linear transformations. In German-language texts the term Matritze is often encountered as a variant spelling alongside Matrize and Matrix.

Notation and basic operations: A Matritze is usually written as A or M and can be of size m×n. Operations include addition and scalar multiplication, and multiplication by another Matritze (when dimensions permit). The transpose A^T switches rows and columns. For square Matritzen, one can compute the determinant det(A) and, if det(A) ≠ 0, the inverse A^{-1}. The rank measures the dimension of the row or column space.

Special types and canonical forms: A diagonal matrix has nonzero entries only on its main diagonal; a symmetric matrix satisfies A = A^T; an orthogonal matrix satisfies A^T A = I. Common decompositions include LU, QR, and the singular value decomposition (SVD), which expresses A as UΣV^T.

Applications: Matritzen provide compact representations of systems of linear equations and linear transformations. They are central in computer graphics for geometric transforms, in statistics for covariance matrices, in machine learning for data representation and linear regression, and in physics and engineering for modeling and simulations.

History and etymology: The matrix concept emerged in 19th-century linear algebra and was formalized by mathematicians such as Gauss and Cayley. The term matrix derives from Latin matrix, meaning a womb or source; Matritze is a spelling variant found in German-language literature.