Let \(M_n(F)\) be the space of \(n \times n\) matrices over a field \(F\).
Prove that the sum and scalar multiples of symmetric matrices are symmetric (so symmetric matrices form a subspace of \(M_n(F)\)).
Prove that the sum and scalar multiples of skew-symmetric matrices are skew-symmetric (so skew-symmetric matrices form a subspace of \(M_n(F)\)).
If \(\operatorname{char}(F) \neq 2\), prove that \(M_n(F) = \operatorname{Sym}_n(F) \oplus \operatorname{Skew}_n(F)\) as an internal direct sum.
Solution
Let \(\operatorname{Sym}_n(F)=\{A:A^t=A\}\) and \(\operatorname{Skew}_n(F)=\{A:A^t=-A\}\).
1. \(\operatorname{Sym}_n(F)\) and \(\operatorname{Skew}_n(F)\) are subspaces of \(M_n(F)\).
Both contain \(0\). For \(A,B\) in one of them, \(c\in F\), and \(\varepsilon=\pm1\) with \(A^t=\varepsilon A\), \(B^t=\varepsilon B\), linearity of the transpose gives \((cA+B)^t=cA^t+B^t=\varepsilon(cA+B)\).
2. If \(\operatorname{char}(F)\neq2\), then \(M_n(F)=\operatorname{Sym}_n(F)+\operatorname{Skew}_n(F)\).
For \(M\in M_n(F)\), \(M=\frac{M+M^t}2+\frac{M-M^t}2\), and \((M^t)^t=M\) shows that the first summand is symmetric and the second skew-symmetric.
3. If \(\operatorname{char}(F)\neq2\), then \(\operatorname{Sym}_n(F)\cap\operatorname{Skew}_n(F)=0\).
If \(X=X^t=-X\), then \(2X=0\), and \(2\) is invertible in \(F\), so \(X=0\).
4. Q.E.D.
Step 1 gives parts 1 and 2, and steps 2 and 3 give part 3.