induced by . Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other. Properties Induced partial ordering For arbitrary square matrices , we write if i.e., is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering . The ordering is called the Loewner order. Inverse of positive definite matrix Every positive definite matrix is invertible and its inverse is also positive definite. If then . Moreover, by the min-max theorem, the kth largest eigenvalue of is greater than or equal to the kth largest eigenvalue of . Scaling If is positive definite and is a real number, then is positive definite. Addition If and are positive-definite, then the sum is also positive-definite. If and are positive-semidefinite, then the sum is also positive-semidefinite. If is positive-definite and is positive-semidefinite, then the sum is also positive-definite. Multiplication If and are positive definite, then the products and are also positive definite. If , then is also positive definite. If is positive semidefinite, then is positive semidefinite for any (possibly rectangular) matrix . If is positive definite and has full column rank, then is positive definite. Trace The diagonal entries of a positive-semidefinite matrix are real and non-negative. As a consequence the trace, . Furthermore, since every principal sub-matrix (in particular, 2-by-2) is positive semidefinite, and thus, when , An Hermitian matrix is positive definite if it satisfies the following trace inequalities: Another important result is that for any and positive-semidefinite matrices, . This follows by writing . The matrix is positive-semidefinite and thus has non-negative eigenvalues, whose sum, the trace, is therefore also non-negative. Hadamard product If , although is not necessary positive semidefinite, the Hadamard product is, (this result is often called the Schur product theorem). Regarding the Hadamard product of two positive semidefinite matrices , , there are two notable inequalities: Oppenheim's inequality: . Kronecker product If , although is not necessary positive semidefinite, the Kronecker product . Frobenius product If , although is not necessary positive semidefinite, the Frobenius inner product (Lancaster–Tismenetsky, The Theory of Matrices, p. 218). Convexity The set of positive