Log-Determinant Function and Properties

The log-determinant function is a function from the set of symmetric matrices in \(\mathbf{R}^{n \times n}\), with domain the set of positive definite matrices, and with values

\[ f(X) = \left\{ \begin{array}{ll} \log \det X & \mbox{if } X \succ 0, \\ +\infty & \mbox{otherwise.} \end{array} \right. \]

The function can be expressed in terms of the (real, positive) eigenvalues of the argument matrix \(X\); it does not depend on its eigenvectors.

This function provides a measure of the volume of an ellipsoid. Precisely, the volume of the ellipsoid

\[ \mathbf{E} = \left\{ x  :  x^TX^{-1}x \le 1 \right\} \]

is given by \(\mbox{\bf vol}(\mathbf{E}) = C_n \prod_{i=1}^n \sqrt{\lambda_i(X)}\), where \(C_n\) is a constant (given by the volume of the unit sphere in \(\mathbf{R}^n\)). Thus, \(\log \mbox{\bf vol}(\mathbf{E}) = \frac{1}{2} f(X) + \mbox{constant}\).

This means that the volume of the ellipsoid is a function of the product of the eigenvalues of the matrix \(X\).

Proof of the concavity of the log-determinant function: We use the fact that a function is convex if and only if its restriction to a line is.