9 6 3 Convexity Of Log Det K

We want to prove the convexity of the map $K\mapsto -\log (det(K))$ over the set of symmetric positive definite matrices. Let us consider $K$ and $S$ two positive definite matrices and some $\lambda \in [0,1]$.

Since $K^{-1/2}SK^{-1/2}$ is symmetric, there exist an orthogonal matrix $U$ and a diagonal matrix $D$, such that $K^{-1/2}SK^{-1/2} =UDU^T$. We set $Q = K^{1/2}U$.

1. Using that $K^{1/2}$ is symmetric and that $UU^T$=I$, it holds

(1)

\begin{equation} QQ^T = K^{1/2}U (K^{1/2}U)^T = K^{1/2}UU^TK^{1/2}=K. \end{equation}

Moreover, by definition of $U$ and $D$ we have

(2)

\begin{equation} QDQ^T = K^{1/2}U DU^TK^{1/2}U= K^{1/2}K^{-1/2}SK^{-1/2}K^{1/2}=S. \end{equation}

2. Using the properties of the logarithm and the det, we have

(3)

\begin{align} -\log(det(\lambda S + (1-\lambda) K)) &= -\log (det(K^{1/2}[\lambda K^{-1/2}SK^{-1/2} + (1-\lambda) I ] K^{1/2}))\\ &= -\log (det(K^{1/2})) - \log (det(\lambda K^{-1/2}SK^{-1/2} + (1-\lambda) I )) - \log (det( K^{1/2}))\\ &= -\log (det(K)) - \log (det(\lambda UDU^T + (1-\lambda) I ))\\ &= -\log (det(K)) - \log (det(\lambda D+ (1-\lambda) I )) . \end{align}

3. Let us denote $D=diag(d_i, i \in \{1, \dots, n\})$. Then it holds

(4)

\begin{align} -\log(det(\lambda D+ (1-\lambda) I )) &= -\log (\prod_{i=1}^n [\lambda d_i+ (1-\lambda)])\\ &= -\sum_{i=1}^n \log (\lambda d_i+ (1-\lambda))\\ &\leq -\lambda \sum_{i=1}^n \log (d_i)+ (1-\lambda)\log(0)\\ &= -\lambda \sum_{i=1}^n \log (d_i)\\ &= -\lambda \log(det(D)), \end{align}

where the penultimate inequality comes from the convexity of the map $x\mapsto -\log(x)$. Using question 2, we obtain

(5)

\begin{align} -\log(det(\lambda S + (1-\lambda) K)) &\leq -\log (det(K)) - \lambda \log (det(D )) \\ &=-\lambda \log (det(S)) -(1- \lambda) \log (det(K )), \end{align}

where the last equality comes from basic properties of the logarithm and the det, and from question 1.