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Symmetric, skew, and the birth of quadratic forms

Every bilinear form splits cleanly into a symmetric and a skew-symmetric piece. The symmetric piece is the one that gives a quadratic form — the function B(x, x) that measures a single vector against itself.

Splitting any form into two halves

A form is symmetric if B(u, v) = B(v, u) for all u, v — equivalently its matrix satisfies A^T = A. It is alternating (skew) if B(v, v) = 0 for all v, which over most fields forces A^T = -A. Over any field where 2 is invertible, every form decomposes uniquely as the sum of a symmetric and a skew part.

Any matrix A splits as A = S + K with:
   S = (A + A^T) / 2   symmetric   (S^T = S)
   K = (A - A^T) / 2   skew        (K^T = -K)

Example  A = [2, 1; 0, 3] from Guide 1:
   A^T = [2, 0; 1, 3]
   S = (A + A^T)/2 = [2,   0.5; 0.5, 3]
   K = (A - A^T)/2 = [0,   0.5; -0.5, 0]
   check:  S + K = [2, 1; 0, 3] = A         OK
The unique symmetric-plus-skew decomposition of a square matrix.

From a symmetric form to a quadratic form

Given a symmetric bilinear form B, feed it the same vector twice: q(x) = B(x, x). The result is a quadratic form — a homogeneous degree-two polynomial in the coordinates, q(x) = x^T A x with A symmetric. Length-squared ||x||^2 is the quadratic form of the inner product; energy, variance, and conic equations are all quadratic forms in disguise.

The link runs both ways. From the form you get q by restricting to the diagonal; conversely the polarization identity B(u, v) = (1/2)( q(u + v) - q(u) - q(v) ) rebuilds the symmetric form from the quadratic one. So a symmetric bilinear form and its quadratic form carry exactly the same information — two views of one object.

A first taste of the skew world

The skew half is not waste — it has its own rich theory. A nondegenerate alternating form is a symplectic form, the geometry behind Hamiltonian mechanics and phase space. Skew forms behave very differently from symmetric ones: for instance a nondegenerate alternating form can exist only in even dimension, a striking fact you'll revisit in the final guide.