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Transposing reflects a matrix across its main diagonal, turning rows into columns. An m × n matrix becomes n × m, and the trace of a square matrix is unchanged.
Transpose
(Aᵀ)ᵢⱼ = Aⱼᵢ
Product rule
(AB)ᵀ = BᵀAᵀ
[[1, 4], [2, 5], [3, 6]] — a 3 × 2 matrix.
Because the row-column pairing swaps, so the last factor must be applied first for the dimensions to line up.