Quadratic Forms

x^TAx = (Pz)^TA(Pz) = z^T(P^TAP)z = z^TDz = lambda1z1^2 + lambda2z2^2 + .... lambdanzn^2

This means we can express matrices in quadratic form. Lambdas are eigenvalues of A and z1,z2 are linear functions of x1,x2 determined by z = P^Tx. x = Pz.

Positive Definiteness: quadratic form of x^TAx is a positive definiteness if x^TAx > 0 for all non-zero vectors of x. As x^TAx = z^TAz, all eigenvalues of x are positive.

If x^TAx < 0, then it is negative definite. If some vectors are positive and some are negative, then it is indefinite.

The eigenvalues of A are roots of the characteristic equation.

matrix A is a symmetric matrix a,c,c,b

lambda1lambda2 = ab - c^2 lmabda1 + lambda 2 = a+b

if IAI > 0 eigenvalues have the same sign if IAI < 0 indefinite
 * if a > 0 and lambda1,2 > 0 then + definite
 * if a < 0 and lambda 1,2 < 0 then - definite