Positive-definite matrix

A real symmetric matrix is positive definite when its quadratic form is positive for every nonzero vector. Equivalently, all its eigenvalues are positive.

Quadratic form: A homogeneous degree-two polynomial in vector coordinates, commonly written as xᵀAx. The defining condition tests whether this expression stays positive away from zero.
Symmetric matrix: A square matrix equal to its transpose. Symmetry ensures real eigenvalues and makes the quadratic-form definition equivalent to familiar tests.
Hessian matrix: The matrix of second partial derivatives of a scalar-valued function. A positive-definite Hessian at a stationary point establishes a strict local minimum.
Indefinite matrix: A symmetric matrix whose quadratic form takes both positive and negative values. Its mixed signs show what fails when positive definiteness is lost.
Convex function: A function whose value along each line segment does not exceed the corresponding linear interpolation of endpoint values. A positive-definite Hessian throughout a convex domain makes a twice-differentiable function strictly convex.
Eigenvalue: A scalar by which a linear transformation scales one of its eigenvectors. A real symmetric matrix is positive definite exactly when every eigenvalue is positive.
Inner product: A bilinear or sesquilinear form that assigns lengths and angles while satisfying positivity conditions. The expression xᵀAx acts as a weighted squared length when A is positive definite.
Covariance matrix: A matrix whose entries are covariances between pairs of random variables. Covariance matrices are positive semidefinite, and become positive definite when no nonzero linear combination has zero variance.
Negative-definite matrix: A symmetric matrix whose quadratic form is negative for every nonzero vector. Negating a positive-definite matrix produces this opposite sign condition.
Condition number: A measure of how strongly a problem's output can change under small input perturbations. The ratio of largest to smallest eigenvalue measures conditioning for positive-definite systems.

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