A theorem on positive semidefinite forms and eigenvalues
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:
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