Positive-definite Matrix - Characterizations

Characterizations

Let M be an n × n Hermitian matrix. The following properties are equivalent to M being positive definite:

1. All its eigenvalues are positive. Let P−1DP be an eigendecomposition of M, where P is a unitary complex matrix whose rows comprise an orthonormal basis of eigenvectors of M, and D is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix M may be regarded as a diagonal matrix D that has been re-expressed in coordinates of the basis P. In particular, the one-to-one change of variable y = Pz shows that z*Mz is real and positive for any complex vector z if and only if y*Dy is real and positive for any y'; in other words, if D is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of M—is positive.
2. The associated sesquilinear form is an inner product. The sesquilinear form defined by M is the function from to such that for all x and y in . For any complex matrix M, this form is linear in each argument separately. Therefore the form is an inner product on if and only if is real and positive for all nonzero z; that is if and only if M is positive definite. (In fact, every inner product on arises in this fashion from a Hermitian positive definite matrix.)
3. It is the Gram matrix of linearly independent vectors. Let be a list of n linearly independent vectors of some complex vector space with an inner product . It can be verified that the Gram matrix M of those vectors, defined by, is always positive definite. Conversely, if M is positive definite, it has an eigendecomposition P−1DP where P is unitary, D diagonal, and all diagonal elements of D are real and positive. Let be the columns of P, each multiplied by the (real) square root of the corresponding eigenvalue . These vectors are linearly independent, and M is their Gram matrix, under the standard inner product of, namely
4. Its leading principal minors are all positive. The kth leading principal minor of a matrix M is the determinant of its upper left k by k sub-matrix. It turns out that 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. Since the kth leading principal minor of a triangular matrix is the product of its diagonal elements up to row k, Sylvester's criterion is equivalent to checking whether its diagonal elements are all positive. This condition can be checked each time a new row k of the triangular matrix is obtained.
5. It has a Cholesky decomposition. The matrix M is positive definite if and only if there exists an lower triangular matrix, with strictly positive diagonal elements, such that . This factorization is called the Cholesky decomposition of M.