Paper 4, Section I, E

Linear Algebra
Part IB, 2018

Define a quadratic form on a finite dimensional real vector space. What does it mean for a quadratic form to be positive definite?

Find a basis with respect to which the quadratic form

x2+2xy+2y2+2yz+3z2x^{2}+2 x y+2 y^{2}+2 y z+3 z^{2}

is diagonal. Is this quadratic form positive definite?