transpose

Transpose, in general, refers to the operation of flipping or shifting an object to another position or orientation. It has specific meanings in mathematics, music, computing, and cryptography.

In mathematics, the transpose of a matrix A is a new matrix A^T formed by swapping its rows and columns. If A is an m-by-n matrix, then A^T is an n-by-m matrix with element A_{ij} moved to position A_{ji}. Real matrices use the standard transpose; complex matrices often employ the conjugate transpose, A*, which combines transposition with complex conjugation. Key properties include (A^T)^T = A, (AB)^T = B^T A^T, and rank(A^T) = rank(A). A matrix is symmetric when A^T = A and skew-symmetric when A^T = -A. Transpose operations are compatible with matrix multiplication by reversing the order of factors.

In data processing and programming, transposition commonly refers to swapping axes of an array, turning an m-by-n array into an n-by-m array. Many software libraries provide a transpose function or operator for tensors, matrices, and higher-dimensional arrays.

In music, to transpose a piece means to shift all pitches by a fixed interval, preserving intervallic structure while changing the key. Transposition is used to accommodate vocal ranges or instrument limitations, such as moving from C major to D major.

In cryptography, a transposition cipher rearranges the letters of plaintext according to a permutation, as opposed to substitution ciphers which replace symbols.

Originating from Latin transponere (“to place across”), transpose is a versatile concept across disciplines, reflecting the idea of repositioning or re-indexing elements.