38. Transpose

The transpose of a matrix flips it across its main diagonal — rows become columns:

If is , then is .

38.1. Properties

• Involution:
• Sum:
• Scalar:
• Product (order reversed):
• Inverse: (when is invertible)
• Determinant:
• Rank:

38.2. Special cases

• Symmetric:
• Skew-symmetric:
• Orthogonal: , i.e.
• Vector transpose: a column vector becomes a row vector and vice-versa

38.3. Connection to dot product

For column vectors :

(see Dot Product)