The formula
(Aᵀ)ᵢⱼ = Aⱼᵢ
What it means
Transposing reflects a matrix across its main diagonal. It is the simplest matrix operation there is, and it turns up everywhere: in dot products, in least squares, and in the definition of symmetric and orthogonal matrices.
How to do it by hand
- Take the first row and write it as the first column
- Repeat for every row
- An m×n matrix becomes n×m
- The diagonal stays where it is
What is worth knowing
Two properties are worth memorising because they surprise people. Transposing a product reverses the order: (AB)ᵀ = BᵀAᵀ, not AᵀBᵀ. And transposing twice gets you back where you started. A matrix that equals its own transpose is symmetric, which forces it to be square and gives it real eigenvalues — a fact that underpins a great deal of statistics, since covariance matrices are always symmetric. Multiplying any matrix by its own transpose also always produces a symmetric result, which is the trick behind the normal equations in least-squares fitting.