Quadratic forms

Given a vector space V, we will say a function φ: V->r is a quadractic form if:

φ (x) = μ belonging to R

 φ (ax)= a^2 x

φ (x+y) + φ (x-y) = 2φ(x) + 2φ(y)

The matrix form is formed by φ (x)= a11 (x1)^2 + a22 (x2)^2 + a33 (x3)^2 + 2a12 x1x2 + a13 x1x3 + 2a23 x2x3. The vector x is placed in a row, then the matrix is formed by placed a11, a22 y a33 in the diagonal and the rest of them is dividing a12, a13 and a23 by two and put the in their symmetric place; finally, the vector x is placed again as a column matrix.

The classification os a quadratic form is:

• Positive: When φ(x) > 0
• Semidefined positive: When φ(x) ≥ 0
• Semidefined negative: When φ(x) ≤ 0
• Defined negative: When φ(x) < 0
• Undefined: When φ(x) = ?

Basically, when you calculate the diagonal matrix of a quadratic form, you see the values of the eigenvalues. If all of them are positive, it's positive. When they are positive and at least one of them is zero, semidefined positive. The same thing with the negative and semidefined but with negative values. If none of these condictions aren't showed, then it's undefined. The quadratic forms are the best way to calculate the positive property of a scalar product, where you need to have a defined positive.