If is positive definite, then the diagonal of is positive and the Cholesky decomposition is unique. Conversely if is lower triangular with nonnegative diagonal then is positive semidefinite. The Cholesky decomposition is especially useful for efficient numerical calculations. A closely related decomposition is the LDL decomposition, , where is diagonal and is lower unitriangular. Other characterizations Let be an real symmetric matrix, and let be the "unit ball" defined by . Then we have the following is a solid slab sandwiched between . if and only if is an ellipsoid, or an ellipsoidal cylinder. if and only if is bounded, that is, it is an ellipsoid. If , then if and only if ; if and only if . If , then for all if and only if . So, since the polar dual of an ellipsoid is also an ellipsoid with the same principal axes, with inverse lengths, we have That is, if is positive-definite, then for all if and only if Let be an Hermitian matrix. The following properties are equivalent to being positive definite: The associated sesquilinear form is an inner product The sesquilinear form defined by is the function from to such that for all and in , where is the conjugate transpose of . For any complex matrix , this form is linear in and semilinear in . Therefore, the form is an inner product on if and only if is real and positive for all nonzero ; that is if and only if is positive definite. (In fact, every inner product on arises in this fashion from a Hermitian positive definite matrix.) Its leading principal minors are all positive The kth leading principal minor of a matrix is the determinant of its upper-left sub-matrix. It turns out that a matrix is positive definite if and only if all these determinants are positive. This condition is known as Sylvester's criterion, and provides an efficient test of positive definiteness of a symmetric real matrix. Namely, the matrix is reduced to an upper triangular matrix by using elementary row operations, as in the first part of the Gaussian elimination method, taking care to preserve the sign of its determinant during pivoting process. Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row , Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. This