3.4 Transpose of a Matrix

3.4 Transpose of a Matrix

The transpose of a matrix $A$, denoted $A^T$ or $A'$, is obtained by interchanging rows and columns:

$$(A^T)_{ij} = a_{ji}$$

Properties

• $(A^T)^T = A$
• $(A + B)^T = A^T + B^T$
• $(kA)^T = kA^T$
• $(AB)^T = B^T A^T$

Symmetric and Skew-Symmetric Matrices

• Symmetric: $A^T = A$ (i.e., $a_{ij} = a_{ji}$)
• Skew-symmetric: $A^T = -A$ (i.e., $a_{ij} = -a_{ji}$, diagonal elements are $0$)

Theorem: Every square matrix $A$ can be uniquely expressed as:

$$A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T)$$

where the first term is symmetric and the second is skew-symmetric.