Atranspose

Atranspose refers to the transpose operation on a matrix, commonly denoted as A^T or A^t. If A is an m-by-n matrix, then A^T is an n-by-m matrix whose entries satisfy (A^T)_{ij} = A_{ji}. In other words, rows of A become columns of A^T and vice versa.

Key properties include that the transpose of a transpose returns the original matrix, (A^T)^T = A. The transpose interacts with matrix multiplication via (AB)^T = B^T A^T. For invertible A, (A^{-1})^T = (A^T)^{-1}. Scalars commute with transpose, so (cA)^T = cA^T for any scalar c. In real matrices, tr(A^T) = tr(A) and det(A^T) = det(A). The product A^T A is always symmetric and, when A has full column rank, positive semidefinite. In the complex case, the conjugate transpose (or Hermitian transpose) A^H is more commonly used, combining transpose with complex conjugation.

Special cases include symmetric matrices, where A^T = A, and skew-symmetric matrices, where A^T = -A. When A is real, A^T and A have the same eigenvalues, though the eigenvectors generally differ; for general complex matrices, A^H provides the analogous Hermitian relation.

Example: If A = [[1, 2, 3], [4, 5, 6]], then A^T = [[1, 4], [2, 5], [3, 6]].

Applications span linear systems, least squares, and geometric interpretations of inner products, since A^T often expresses projections, coordinate changes, and Gram matrices in various computations.