is not a total order, however, as in general may be indefinite. A common alternative notation is , , and for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way. Examples Eigenvalues Let be an Hermitian matrix (this includes real symmetric matrices). All eigenvalues of are real, and their sign characterize its definiteness: is positive definite if and only if all of its eigenvalues are positive. is positive semi-definite if and only if all of its eigenvalues are non-negative. is negative definite if and only if all of its eigenvalues are negative is negative semi-definite if and only if all of its eigenvalues are non-positive. is indefinite if and only if it has both positive and negative eigenvalues. Let be an eigendecomposition of , where is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of , and is a real diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix may be regarded as a diagonal matrix that has been re-expressed in coordinates of the (eigenvectors) basis . Put differently, applying to some vector , giving , is the same as changing the basis to the eigenvector coordinate system using , giving , applying the stretching transformation to the result, giving , and then changing the basis back using , giving . With this in mind, the one-to-one change of variable shows that is real and positive for any complex vector if and only if is real and positive for any ; in other words, if is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of —is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix is available. Decomposition Let be an Hermitian matrix. is positive semidefinite if and only if it can be decomposed as a product of a matrix with its conjugate transpose. When is real, can be real as well and the decomposition can be written as is positive definite if and only if such a decomposition exists with invertible. More generally, is positive semidefinite with rank if and only if a decomposition exists