Sylvester's criterion

Sylvester's criterion states that a Hermitian matrix is positive definite exactly when all its leading principal minors are positive.

Positive-definite matrix: A Hermitian matrix whose quadratic form is positive for every nonzero vector. This is the matrix property the determinant tests characterize.
Hermitian matrix: A complex square matrix equal to its conjugate transpose. The criterion applies to Hermitian matrices, including real symmetric matrices.
Hessian matrix: The square matrix of second partial derivatives of a scalar function. A positive-definite Hessian certifies a strict local minimum, and Sylvester's criterion tests it.
Positive-semidefinite matrix: A Hermitian matrix whose quadratic form is nonnegative for every vector. Semidefiniteness allows zero eigenvalues, so strict positivity of all leading minors is not its test.
Principal minor: The determinant of a submatrix formed by selecting the same index set for rows and columns. The criterion checks principal minors from the first rows and columns.
Determinant: A scalar associated with a square matrix that encodes volume scaling and whether the matrix is singular. Each condition in the criterion is a sign test on a determinant.
Second derivative test: A method for classifying a stationary point using the second derivatives of a function. Its multivariable form often reduces to testing the Hessian with this criterion.
Sylvester's law of inertia: The theorem that an Hermitian matrix's counts of positive, negative, and zero eigenvalues are invariant under congruence. It concerns inertia under transformations, not a positivity test from leading determinants.
Leading principal minor: A principal minor formed from the first k rows and columns of a matrix, for some k. These are the specific minors whose positivity suffices.
Matrix minor: The determinant of a square submatrix selected from a matrix. Principal minors are a particular kind of matrix minor.

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