Transposing reflects a matrix across its main diagonal: the entry at row i, column j moves to row j, column i. Press Transpose A and the flipped matrix appears. It is the cheapest operation on this site and one of the most used, because so many formulas are stated in terms of it.
What transposing does
An m×n matrix becomes n×m, so the operation is defined for every shape — no squareness required. A column vector becomes a row vector, which is why the notation xᵀy is the standard way to write a dot product.
Three identities are worth committing to memory, because they come up constantly and two of them catch people out. Transposing twice returns the original: (Aᵀ)ᵀ = A. Transposing a sum distributes: (A + B)ᵀ = Aᵀ + Bᵀ. But transposing a product reverses it: (A·B)ᵀ = Bᵀ·Aᵀ, not AᵀBᵀ. The reversal is forced by the shapes — if A is 2×3 and B is 3×4, then Aᵀ is 3×2 and Bᵀ is 4×3, and only Bᵀ·Aᵀ has dimensions that agree.
Worked example
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 1 | 4 |
| 2 | 5 |
| 3 | 6 |
The first row became the first column. A matrix equal to its own transpose is called symmetric, and symmetric matrices are unusually well behaved: their eigenvalues are always real, and they are the ones Cholesky decomposition applies to.
Common mistakes
- Writing (A·B)ᵀ = Aᵀ·Bᵀ. The order reverses. This is the single most common slip involving transposes.
- Expecting the diagonal to move. Entries where i = j stay exactly where they are; only the off-diagonal ones swap.
- Assuming the shape is preserved. A 2×5 matrix transposes to 5×2, which may break the multiplication you were about to do.
- Confusing transpose with inverse. They coincide only for orthogonal matrices, where Aᵀ = A⁻¹. In general they are unrelated.