Transpose

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At). It is achieved by any one of the following equivalent actions: In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At). It is achieved by any one of the following equivalent actions: Formally, the i-th row, j-th column element of AT is the j-th row, i-th column element of A: If A is an m × n matrix, then AT is an n × m matrix. To avoid confusing the reader between the transpose operation and a matrix raised to the tth power, the A ⊤ {displaystyle mathbf {A} ^{ op }} symbol denotes the transpose operation. The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. For matrices A, B and scalar c we have the following properties of transpose: A square matrix whose transpose is equal to itself is called a symmetric matrix; that is, A is symmetric if A square matrix whose transpose is equal to its negative is called a skew-symmetric matrix; that is, A is skew-symmetric if A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose); that is, A is Hermitian if A square complex matrix whose transpose is equal to the negation of its complex conjugate is called a skew-Hermitian matrix; that is, A is skew-Hermitian if