This web page gives some elmentary definitions and results concerning quadratic forms. They are needed to help with the proof of Legendre's theorem about numbers that are the sum of 3 squares.
Definition: A quadratic form F is a polynomial in n variables
F(x1,...,xn) = sum(i=1,...,n) sum(j=1,...,n) ai,j xi x_j
Definition: A quadratic form F is positive definite if F(x1,...,xn) > 0 unless all its parameters x1,...,xn are 0.
Definition: A quadratic form F is negative definite if F(x1,...,xn) < 0 unless all its parameters x1,...,xn are 0.
Definition: A quadratic form F represents an integer m if F(x1,...,xn)=m for some integers x1,...,xn.
Definition: The matrix of the above quadratic form F is A = {i,j}
Thus, using matrix and vector notation, F(x) = xAx' Note that the matrix A is symmetrical.
Definition: The determinant D(F) of the above quadratic form F is det A.
Definition: A square matrix is unimodular if its coefficients are integers and its determinant is 1.
Definition: Let F, G be quadratic forms with respective matrices A, B. Then F and G are equivalent, notated F ~ G, if there is a unimodular matrix T such that
B = TAT−1
Only ternary quadratic forms will be needed in this proof. Simpler definitions and notation can thus be used:
Definition: A ternary quadratic form is a quadratic polynomial in 3 variables
F(x, y, z) = ax2 + by2 + cz2 + 2dyz + 2exz + 2fxy
Definition: The determinant D(F) of a ternary quadratic form F is
| D(F) = |
|
= abc + 2def - ad2 - be2 - cf2 |