= Symmetric and alternating parts of a bilinear form
{title2=$B=S+K$}
Over a <field> whose <characteristic> is not two, a <bilinear form> has a unique decomposition into a <symmetric bilinear form> $S$ and an <alternating bilinear form> $K$:
$$
S(v,w)=\frac{B(v,w)+B(w,v)}2,\qquad K(v,w)=\frac{B(v,w)-B(w,v)}2.
$$
Adding and subtracting the equations $B(v,w)=S(v,w)+K(v,w)$ and $B(w,v)=S(v,w)-K(v,w)$ both proves uniqueness and gives these formulas. In <matrix> notation, $S$ has <matrix> $(A+A^T)/2$ and $K$ has <skew-symmetric matrix> $(A-A^T)/2$.<Bilinear Forms by Keith Conrad>[Theorem 1.7: symmetric and alternating decomposition]{ref}
The diagonal values $q(v)=B(v,v)$ alone do not determine a general <bilinear form>, because $K(v,v)=0$ for every <vector>. For the <matrix of a bilinear form> in the example above,
$$
A=\underbrace{\begin{pmatrix}1&1\\1&1\end{pmatrix}}_{\text{symmetric part}}+\underbrace{\begin{pmatrix}0&1\\-1&0\end{pmatrix}}_{\text{alternating part}},\qquad q(x,y)=(x+y)^2.
$$
Thus two different <bilinear forms>, represented by $A$ and its <symmetric matrix> part, have exactly the same <quadratic form>. To recover a <bilinear form> from its diagonal, one needs extra information such as the hypothesis that it is a <symmetric bilinear form>.
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