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 A by producing another matrix, often denoted by AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley. In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation RT. The transpose of a matrix A, denoted by AT, ^⊤A, A^⊤, , A′, Atr, tA or At, may be constructed by any one of the following methods: Reflect A over its main diagonal (which runs from top-left to bottom-right) to obtain AT Write the rows of A as the columns of AT Write the columns of A as the rows of AT 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. In the case of square matrices, AT may also denote the Tth power of the matrix A. For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as TA. An advantage of this notation is that no parentheses are needed when exponents are involved: as (^TA)^n = ^T(A^n), notation ^TA^n is not ambiguous. In this article this confusion is avoided by never using the symbol T as a variable name. 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 A square matrix whose transpose is equal to its inverse is called an orthogonal matrix; that is, A is orthogonal if A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix; that is, A is unitary if Let A and B be matrices and c be a scalar.