rankAT

rankAT refers to the rank of the transpose of a matrix A, written as rank(A^T). In linear algebra, transposing a matrix swaps its rows and columns, but does not change the underlying linear independence of the rows or columns. Consequently, rank(A^T) equals rank(A); the row rank and the column rank are both equal to the rank of A.

For an m-by-n matrix A, rank(A^T) is at most min(n, m), and in fact rank(A^T) = rank(A). This common value r is equal to the number of nonzero singular values of A and also to the dimensions of both the row space and the column space of A. The equality of rank(A) and rank(A^T) reflects the fact that row rank and column rank are the same invariant of a matrix.

Computationally, rankAT can be determined by performing Gaussian elimination (row reduction) on A^T or A; both procedures yield the same rank. In numerical linear algebra, the rank is often taken as the number of singular values of A that exceed a chosen tolerance, which defines the numerical rank.

Example: Let A be the 2-by-3 matrix [[1, 2, 0], [0, 0, 3]]. A has two rows that are linearly independent, so rank(A) = 2. The transpose A^T is 3-by-2 and also has rank 2, so rankAT = 2.

Applications of rankAT include assessing solvability of linear systems, understanding independence of rows and columns, and informing algorithms in areas such as data compression, cryptography, and machine learning where transpose operations are common. See also: matrix rank, row rank, column rank, singular value decomposition.