Seja $V$ o espaço vetorial das matrizes quadradas $n\ x\ n$, $U$ o subespaço das matrizes simétricas e $W$ o subespaço das matrizes antissimétricas, $V = U \oplus W$.

$M \in U\ \Rightarrow\ M = M^t$

$M \in W\ \Rightarrow\ M = -M^t$

Seja $A \in V$, $A = \dfrac{1}{2}(A + A^t) + \dfrac{1}{2}(A - A^t)\ {\large (I)}$.

$(A + A^t)^t = A^t + A\ \Rightarrow\ (A + A^t) \in U\ {\large (II)}$

$(A - A^t)^t = -(A - A^t)\ \Rightarrow\ (A - A^t) \in W\ {\large (III)}$

${\large (I)}\ \wedge\ {\large (II)}\ \wedge\ {\large (III)}\ \Rightarrow\ V = U + W\ {\large (IV)}$

Seja $M \in U\ \wedge\ M \in W$:

$M = M^t\ \wedge\ -M = M^t\ \Rightarrow\ M = -M\ \Rightarrow\ M = O\ \Rightarrow\ U \cap W = \{O\}\ {\large (V)}$.

${\large (IV)}\ \wedge\ {\large (V)}\ \Rightarrow\ V = U \oplus W$

Quod Erat Demonstrandum.