Positive Definite Matrix Checker

About this calculator

For a 2×2 symmetric matrix written as [[a, b], [b, c]], its definiteness — whether the quadratic form it represents is always positive, always negative, or mixed — can be read directly from two numbers: the leading entry a and the determinant, det = ac − b². By Sylvester’s criterion applied to the 2×2 case, the matrix is positive definite if a is greater than 0 and the determinant is greater than 0; positive semidefinite if a is at least 0 and the determinant is at least 0 (with equality allowed); negative definite if a is less than 0 and the determinant is greater than 0; and indefinite whenever the determinant is negative, meaning the matrix produces both positive and negative values depending on direction.

This classification matters across linear algebra and its applications: in optimization, the Hessian matrix of a function at a critical point must be positive definite for that point to be a local minimum (negative definite for a local maximum, indefinite for a saddle point); in statistics, a valid covariance matrix must be positive semidefinite; and in numerical methods, positive definiteness guarantees certain matrix decompositions (like the Cholesky decomposition) exist. This calculator takes the entries a, b, and c of a symmetric 2×2 matrix and classifies it using the leading minor and determinant.