Check whether a symmetric matrix is positive definite using Sylvester's criterion.
How It Works
How Positive Definite Matrix Checker Works
For a symmetric matrix, Sylvester's criterion says the matrix is positive definite exactly when every leading principal minor (the determinant of the top-left k×k block, for every k) is positive. The calculator computes each of these minors and checks that all of them come out greater than zero.
FAQ
Frequently Asked Questions
Why does the matrix need to be symmetric first?
Positive definiteness is a concept defined for symmetric matrices — for a non-symmetric matrix, the usual test based on eigenvalues or leading minors doesn't directly apply in the same way, so the calculator checks symmetry before running the test.
Where does positive definiteness come up in practice?
It's central to optimization (a positive definite Hessian confirms you've found a true minimum, not a saddle point) and to statistics, where a valid covariance matrix must be positive semi-definite.
