A symmetric matrix is classified as positive definite if and only if its quadratic form yields a strictly positive scalar for every non-zero vector, which geometrically corresponds to the Hessian matrix at a critical point defining a local minimum with upward-opening curvature (a paraboloid). This concept integrates linear algebraic properties—specifically that all eigenvalues are positive, leading principal minors are positive, and pivots from Gaussian elimination are positive—with multivariable calculus criteria for optimization. The theory establishes an equivalence between the sum-of-squares decomposition of a quadratic form via completing the square (related to LDL factorization) and the spectral diagonalization of symmetric matrices, confirming that such functions lack saddle points in any direction.
A symmetric matrix is classified as positive definite if and only if its quadratic form yields a strictly positive scalar for every non-zero vector, which geometrically corresponds to the Hessian mat…