Transposed

Transposed is the past participle of transpose, meaning altered by transposition. It describes something that has been rearranged by swapping its structure along a fixed reference. The term is used in several fields.

In mathematics and linear algebra, a transpose is an operation on a matrix or array that flips it over its diagonal. The transpose of a matrix A, denoted A^T, has entries a_{ij} in position (j,i). This converts rows to columns. For example, a 2x3 matrix becomes a 3x2 matrix. Key properties include (A^T)^T = A and (AB)^T = B^T A^T; the rank is preserved. For complex matrices the conjugate transpose (also called the Hermitian transpose) is denoted A^H or A*, combining transposition with complex conjugation.

In music, transposition means shifting every note of a piece up or down by a fixed interval, or writing for an instrument in a different key. A piece written in D major, if transposed up a whole step, becomes E major. Transposing instruments are designed to sound at a pitch different from written.

In data analysis and computing, transposed data means swapping rows and columns, turning a matrix or table on its side. This is common when converting between formats or preparing data for algorithms that expect a particular orientation.

Transposed thus denotes either the result of a transposition operation or the property of having been subject to transposition, across domains with precise, context-dependent meanings.