The transpose

The transpose Aᵀ mirrors a matrix across its main diagonal, the line running down and to the right from the top-left corner: (Aᵀ)ᵢ,ⱼ = Aⱼ,ᵢ. Rows become columns, so an m × n matrix becomes n × m.

A vector is a matrix with one column, so its transpose is a matrix with one row. That is why texts often write a column vector inline as x = [x₁, x₂, x₃]ᵀ. A scalar is its own transpose: a = aᵀ.

Transposing a product reverses the order: (AB)ᵀ = BᵀAᵀ.