transponiert

Transponiert is the result of applying the transpose operation to a matrix or array. In mathematics, to transpose means to flip a matrix over its diagonal, turning rows into columns and columns into rows. For an m-by-n matrix A, its transpose A^T (also written A^t) is an n-by-m matrix with entries (A^T)_{ij} = A_{ji}. A transposed row becomes a column, and a transposed column becomes a row. The term transponieren is the German verb for this operation, and die transponierte Matrix refers to the transposed matrix.

Notation and interpretation: The transpose is typically denoted A^T or A^t. In practice, A^T is the matrix obtained by swapping indices i and j. For example, if A = [ [1, 2, 3], [4, 5, 6] ], then A^T = [ [1, 4], [2, 5], [3, 6] ].

Properties: The transpose operation is involutive: (A^T)^T = A. It distributes over addition and scalar multiplication: (A + B)^T = A^T + B^T, and (cA)^T = c A^T for any scalar c. It reverses the order of multiplication: (AB)^T = B^T A^T, where the inner dimensions must align. For complex matrices, the conjugate transpose (also called the Hermitian transpose) is denoted A^H or A*, which combines transposition with complex conjugation.

Special cases and terminology: If A is real, A^T is the same as the transpose without any conjugation. A matrix is called symmetric when A^T = A, and skew-symmetric when A^T = -A.

Applications: Transpose is fundamental in linear algebra, appearing in solving linear systems, forming symmetric matrices, changing basis, and expressing dot products in matrix form. It underpins many algorithms in graphics, statistics, and numerical analysis.