= Symmetric bilinear form
{title2=$B(v,w)=B(w,v)$}
{wiki}
A <bilinear form> $B$ is symmetric when $B(v,w)=B(w,v)$ for every pair of <vectors> $v,w$. Equivalently, its <matrix of a bilinear form> in any <basis> is a <symmetric matrix>, since $A_{ij}=B(e_i,e_j)=B(e_j,e_i)=A_{ji}$. Symmetry makes the relation $B(v,w)=0$ reciprocal, giving the <bilinear orthogonal complement> a geometric interpretation.
Over the <real numbers>, set $q(v)=B(v,v)$. The <polarization identity>
$$
B(v,w)=\frac12\bigl(q(v+w)-q(v)-q(w)\bigr)
$$
recovers the <symmetric bilinear form> from its <quadratic form>: expand $q(v+w)$ and use symmetry to combine the two cross terms. For $q(x,y)=x^2-y^2$, take $v=(2,1)$ and $w=(1,3)$. Then $q(v+w)=-7$, $q(v)=3$, and $q(w)=-8$, so $B(v,w)=(-7-3+8)/2=-1=2\cdot1-1\cdot3$. The same formula works whenever the <characteristic> is not two.https://www.ma.imperial.ac.uk/~anskor/LINEAR.PDF[Skorobogatov, Linear Algebra and Matrices, p. 12: recovering a symmetric form from the diagonal]{ref}
Three real <diagonal matrices> show how much variety symmetry allows. The <identity matrix> $\operatorname{diag}(1,1)$ gives the <positive-definite bilinear form> $q=x^2+y^2$. The <matrix> $\operatorname{diag}(1,-1)$ gives $q=x^2-y^2$ and is <nondegenerate> despite the <isotropic vector> $(1,1)$. The <matrix> $\operatorname{diag}(1,0)$ gives $q=x^2$ and is a <degenerate bilinear form> with <radical of a bilinear form> $\operatorname{span}\{(0,1)\}$. Their <level sets> make the distinction visible: a <circle>, two branches of a <hyperbola>, and two parallel lines for $q=1$.
\Image[/quadratic-form-level-sets.png]
{title=Unit and zero level sets distinguish positive, indefinite nondegenerate, and degenerate symmetric forms}
The <inertia of a bilinear form> records the numbers of positive, negative, and zero directions after <matrix congruence> diagonalization. By <Sylvester's law of inertia>, these counts survive <change of basis>; the three examples have <inertia of a bilinear form> $(2,0,0)$, $(1,1,0)$, and $(1,0,1)$. Historically, Sylvester stated this invariance in an 1852 paper on reducing real <quadratic forms> to sums of positive and negative squares. Ghys and Ranicki reproduce his statement and distinguish it from his subsequent 1853 proof.https://arxiv.org/pdf/1512.09258[Étienne Ghys and Andrew Ranicki, Signatures in algebra, topology and dynamics, §1.2: Sylvester's statement and proof]{ref}
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