Concept guide
Real inner-product space: Sylvester conditions as a universal positive-definiteness certificate for a matrix form on ℝ²
Let M be the symmetric integer matrix with first row (2,1) and second row (1,3), and ⟨x,y⟩_M=xᵀMy for x,y∈ℝ². (1) Prove separately that ⟨·,·⟩_M is bilinear, symmetric and positive definite, hence an inner product on ℝ²; positive definiteness is certified universally by the 2-by-2 Sylvester conditions, with both sufficiency and necessity proved in full. (2) Decide why the forms given by N (rows (1,2),(2,1)), P (rows (1,1),(1,1)) and R (rows (2,1),(0,3)) are not inner products.