| # A theorem on positive semidefinite forms and eigenvalues Theorem: (Link with SED) A quadratic form , with is non-negative (resp. positive-definite) if and only if every eigenvalue of the symmetric matrix is non-negative (resp. positive). Proof: Let be the SED of . If , then gor every . Thus, for every : Conversely, if there exist for which , then choosing will result in . Likewise, a matrix is PD if and only if is a positive-definite function, that is, for every , and if and only if . When for every , then the condition trivially implies for every , which can be written as . Since is orthogonal, it is invertible, and we conclude that . Conversely, if for some , we can achieve for some non-zero . See also: