transposes

Transposes refers to several related operations that rearrange structure by swapping orientation or shifting pitch. In mathematics and music, the term describes distinct but analogous ideas: interchanging axes in a matrix or shifting every note by a fixed interval.

In linear algebra, the transpose of a matrix A, denoted A^T, is formed by interchanging its rows and columns. If A is an m×n matrix, then A^T is an n×m matrix with entries A^Tij = Aji. For complex matrices, the conjugate transpose (or Hermitian transpose) A* combines transposition with complex conjugation. Core rules include (A + B)^T = A^T + B^T and (AB)^T = B^T A^T, with (A^T)^T = A. Transpose operations preserve many properties, such as rank and determinant (for square A, det(A^T) = det(A)), and they relate to dual spaces in linear theory.

In data representation, transposing a table or matrix means switching its rows and columns, often to switch between formats that are row-oriented or column-oriented for analysis or computation.

In music, transposition shifts every pitch in a piece by a constant interval, producing a different key while preserving musical relationships. Transposition is common for instrument transposition, vocal ranges, or adapting music to a different vocal or instrumental range, and it distinguishes concert pitch from written pitch.

Across disciplines, transposes provide a reversible rearrangement that reorients structure while preserving core relationships.