Bilinear forms pair two vectors to produce a scalar. Quadratic forms measure a vector against itself. Together they describe geometry that may not come from a fixed Euclidean inner product.
Bilinear and sesquilinear forms
Definition. A bilinear form on a vector space \(V\) over \(F\) is a function \(B:V\times V\to F\) that is linear in each variable. Over \(\mathbb C\), one often uses sesquilinear forms, linear in one variable and conjugate-linear in the other. A bilinear form is symmetric if \(B(u,v)=B(v,u)\) and alternating if \(B(v,v)=0\) for all \(v\).
Matrix representation and congruence
Proposition. If \(B=(e_1, \ldots,e_n)\) is a basis of \(V\), a bilinear form \(\beta\) has Gram matrix \(G=(\beta(e_i,e_j))\), and \[ \beta(x,y)=[x]_B^\top G [y]_B. \] If \(P \) is a change-of-basis matrix, the Gram matrix changes by congruence: \[ G\mapsto P^\top GP. \]
Proof. Expand \(x=\sum_i x_ie_i\) and \(y=\sum_j y_je_j\). Bilinearity gives \(\beta(x,y)=\sum_{i,j}x_iy_j\beta(e_i,e_j)\). If the new basis vectors are columns of \(P\) in the old coordinates, substituting their coordinate vectors into the formula gives \(P^\top GP\).
Quadratic forms and polarization
Definition. A quadratic form associated with a symmetric bilinear form is \(q(v)=\beta(v,v)\). If the field has characteristic not equal to \(2\), the symmetric bilinear form is recovered by polarization: \[ \beta(u,v)=\frac{1}{2}\big(q(u+v)-q(u)-q(v)\big). \]
Diagonalization by congruence
Theorem. Over a field of characteristic not equal to \(2\), every symmetric bilinear form on a finite-dimensional space has a basis in which its matrix is diagonal. Equivalently, every quadratic form can be written as \[ q(x)=a_1x_1^2+ \cdots+a_rx_r^2 \] in suitable coordinates.
Proof. If some \(v\) has \(q(v)\ne0\), split \(V=Fv\oplus (Fv)^\perp\) and proceed by induction on the orthogonal complement. If \(q(v)=0\) for all \(v\), then polarization gives the zero form. This recursive orthogonalization diagonalizes the form.
Sylvester law of inertia
Theorem. Over \(\mathbb R\), every symmetric matrix is congruent to a diagonal matrix with entries \(1\), \(-1\), and \(0\). The numbers of positive, negative, and zero entries are invariants of the quadratic form. They are called the inertia.
Proof. Diagonalize the form by congruence and rescale nonzero diagonal entries to \(\pm1\). The number of positive directions is the largest dimension of a subspace on which \(q\) is positive definite; the number of negative directions is analogous. These characterizations are coordinate-free, so the counts are invariant.
Symplectic forms
Example. The standard alternating form on \(F^{2n}\) has matrix \[ J=\begin{pmatrix}0&I\\-I&0\end{pmatrix}. \] It is nondegenerate but not symmetric. Its geometry is different from inner-product geometry: every vector satisfies \(\beta(v,v)=0\), so length is not the relevant concept.
Remark. Similarity \(P^{-1}AP\) changes the matrix of an operator. Congruence \(P^\top GP\) changes the matrix of a form. Confusing the two is one of the most common mistakes in this topic.