Pure-quadratic polynomial/R/Change of variables/Pure form/Fact

Principal axis theorem for pure-quadratic polynomials

Every real pure-quadratic polynomial

F = \sum_{i \leq j} a_{i j} X_{i} X_{j}

has, with respect to a suitable

orthonormal basis (with respect to the standard inner product) the form

F = \sum_{1 \leq i \leq n} r_{i} U_{i}^{2} .

The $r_{i}$ are the eigenvalues of the square symmetric matrix

M = {(α_{i j})}_{1 \leq i, j \leq n},

where

α_{i j} = {\begin{matrix} a_{i j} for i = j, \\ \frac{a_{i j}}{2} for i < j, \\ \frac{a_{j i}}{2} for i > j . \end{matrix}