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world:logdet
Table of Contents
Concavity of the $\log \det$ function
Let the space of real-valued symmetric positive matrices. We show
Lemma: The function is concave on .
Proof
Let and . Since , it is diagonalisable in some orthonormal basis and write the (possibly repeated) entries of the diagonal. Note in particular that . Then, where the last inequality follows from the concavity of the . Now, rewrite the rhs as:
Derivatives
Lemma: The derivative of the real valued function defined on is given at a which is symmetric positive by: where, for all real valued function defined on , denotes the matrix such that for all , is the partial derivative of with respect to .
Proof
Recall that for all we have where is the -cofactor associated to . For any fixed , the component does not appear in anywhere in the decomposition , except for the term . This implies Recalling the identity so that , we finally get where the last equality follows from the fact that is symmetric.
world/logdet.txt · Last modified: 2022/03/16 07:40 (external edit)
