Matrix Input
Warning: Matrix is not symmetric. Results use eigenvalues of the input as-is. For definiteness, A should equal Aáµ€.
Result
Enter a symmetric matrix and click Check Definiteness
Classify a symmetric matrix as positive definite, negative definite, positive semi-definite, negative semi-definite, or indefinite. Uses eigenvalues and Sylvester's criterion (leading principal minors).
PD ⟺ all leading principal minors > 0
A symmetric matrix A is positive definite iff xᵀAx > 0 for all x ≠0, equivalently iff all eigenvalues are positive, equivalently iff all leading principal minors are positive.
Leading principal minors: Δ₠= aâ‚â‚, Δ₂ = det(top-left 2×2), Δ₃ = det(A) for 3×3.
"Definiteness is crucial for optimization — a positive definite Hessian means a local minimum. Let's connect the algebra to the geometry."