Transponaten

Transponaten are the transposed matrices of a given matrix in linear algebra. The term is the plural form of the German word Transponat, used in many German-language texts to refer to the transpose operation or its results. Transponaten play a basic role in mathematical reasoning and in many applied fields, where changing the orientation of data can facilitate computations and expose structural properties.

For a real or complex matrix A of size m by n, the Transponat A^T (often denoted A^t) is the n by m matrix obtained by swapping rows with columns. In other words, the entry in the i-th row and j-th column of A becomes the entry in the j-th row and i-th column of A^T. If A is square, A^T is also square and shares many algebraic relationships with A.

Key properties of transponaten include: (A^T)^T = A, so the transpose operation is involutive; (AB)^T = B^T A^T, reflecting the reversal of multiplication order; det(A^T) = det(A) for square matrices; A is symmetric when A^T = A, and A is skew-symmetric when A^T = -A. In complex contexts, the conjugate transpose A* combines transposition with complex conjugation and is widely used in inner product spaces.

An example: for A = [[1, 2], [3, 4]], the Transponat A^T = [[1, 3], [2, 4]]. Transponaten are used in statistics to form Gram matrices (A^T A), in computer graphics, in solving linear systems, and in the study of eigenvalues and orthogonality.