# Transpose

The transpose of a matrix is an operator, typically denoted as a superscripted "T," that flips a matrix over its diagonal and switches the row and column indices of the matrix. More specifically (refer to the matrix notation page for a refresher if necessary), the transpose of an m by n matrix, A, is an n by m matrix, denoted A^{T}, such that the i,j^{th} entry of A is the j,i^{th} entry of A^{T}. We denote the i,j^{th} entry of A as a_{i,j} and the j,i^{th} entry of A^{T} as . Using this notation, A^{T} is an n by m matrix defined by the relations

for all i = 1, 2, ..., m and j = 1, 2, ..., n (i runs through the rows of A, of which there are m, and j runs through the columns of A, of which there are n). For example, if A is the following 3 × 5 matrix

then A^{T} is the following 5 × 3 matrix

Note that:

meaning that the transpose of the transpose of A is just A itself.

The transpose of an n by 1 column vector v is the 1 by n row vector v^{T}.

We would usually write v as v = [3, -4, 2]^{T} and v^{T} as v^{T} = [3, -4, 2]. We write v and v^{T} this way because [3, -4, 2] is written horizontally on the page and therefore should be the row vector v^{T}.

See also matrix notation.