ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3749 on D4571: Real matrix
Partition of real square matrix into sum of symmetric and antisymmetric parts
Formulation 0
Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix.
Then
(1) \begin{equation} A = \frac{1}{2} (A + A^T) + \frac{1}{2} (A - A^T) \end{equation}
(2) $\frac{1}{2} (A + A^T)$ is a D399: Symmetric matrix
(3) $\frac{1}{2} (A - A^T)$ is an D5204: Antisymmetric matrix
Formulation 1
Let $A \in \mathbb{R}^{N \times N}$ be a D4571: Real matrix.
Then
(1) \begin{equation} A = \frac{A + A^T}{2} + \frac{A - A^T}{2} \end{equation}
(2) $\frac{A + A^T}{2}$ is a D399: Symmetric matrix
(3) $\frac{A - A^T}{2}$ is an D5204: Antisymmetric matrix