Any matrix has a unique decomposition of a sum of a symmetric and anti symmetric matrix proof

Question

I am given the following task:

Any $n \times n$ matrix $\ A$ can always be written,$$ A= S+ C$$

Where $ S$ is symmetric and $ C$ is antisymmetric.

Prove that this decomposition is unique.

My proof:

Let $S=\frac{1}{2}(A+A^T)$ and $C=\frac{1}{2}(A-A^T)$. Then $S^T=S$, $C=-C^T$. Also, $S+C=A$.

Suppose $A=S'+C'$, where $(S')^T=S'$ and $(C')^T=-C'$. Then $A+A^T=2S'$ and $A-A^T=2C'$. Therefore, $S=S'$ and $C=C'$, so the decomposition is unique.

Is my proof correct?

Answer 1

$A+A^T=2S',A-A'^T=2C'$,Apart from that minor point, your proof is good. Note that it only works over fields of characteristic not equal to 2. or, more generally, over commutative rings with identity in which 1+1 is invertible.