What criteria must be met for a matrix to be classified as positive definite?

A matrix is classified as positive definite if all of its eigenvalues are positive. This criterion is crucial because for a matrix ( A ) to be positive definite, it must satisfy certain properties, particularly in relation to quadratic forms. For any non-zero vector ( x ), the quadratic form ( x^T A x > 0 ) must hold true. This implication leads directly to the requirement that all eigenvalues of the matrix must be positive. The positivity of the eigenvalues guarantees that all associated quadratic forms yield a positive result for any non-zero vector.

In contrast, the other criteria do not align with the definition of positive definiteness. If all eigenvalues were negative, the matrix would be considered negative definite. Zero eigenvalues would indicate that the matrix has a non-trivial null space, leading to a non-positive definite classification. Similarly, having all rows contain non-zero entries does not provide information about the definiteness of the matrix since it's possible for a matrix with all non-zero rows to still have negative or zero eigenvalues. Thus, the requirement for all eigenvalues to be positive stands as the definitive criterion for a matrix to be deemed positive definite.