Flipping a matrix across its diagonal reveals surprisingly deep properties
Take a grid of numbers and mirror it across the line running from top left to bottom right, so rows become columns. That simple move, the transpose, was introduced by Arthur Cayley in 1858, and it quietly underpins ideas from symmetry and rotations to the humble dot product.
Formally, the entry in row i and column j of the transpose equals the entry in row j and column i of the original. A matrix with m rows and n columns therefore becomes one with n rows and m columns. Notation varies widely; besides a superscript T, writers use a leading t, a trailing tr and other marks.
Several important families of square matrices are defined by how they behave under this flip. A symmetric matrix is unchanged by it, while a skew-symmetric one turns into its own negative. An orthogonal matrix is one whose transpose is also its inverse, so flipping it undoes it. With complex numbers, analogous roles are played by Hermitian and unitary matrices, which pair the flip with swapping each entry for its complex conjugate.
The operation obeys tidy rules. Doing it twice returns the original, making it self-inverse, and it respects addition and scaling, so it acts as a linear map between spaces of matrices. Transposing a product, however, reverses the order of the factors, and that reversal extends to chains of any length. A square matrix and its transpose share the same determinant and the same eigenvalues, because they have the same characteristic polynomial. If a matrix is invertible, so is its transpose, and the inverse of one is the transpose of the inverse of the other.
One everyday use: writing one column vector as a row and multiplying it by another column gives a one-entry matrix holding their dot product. And for any real matrix A, multiplying its transpose by A always yields a positive-semidefinite result.
Source: Transpose
