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 AT, such that the i,jth entry of A is the j,ith entry of AT. We denote the i,jth entry of A as ai,j and the j,ith entry of AT as . Using this notation, AT 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 AT 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 vT.

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

See also matrix notation.