| # Log-Determinant Function and Properties The log-determinant function is a function from the set of symmetric matrices in , with domain the set of positive definite matrices, and with values The function can be expressed in terms of the (real, positive) eigenvalues of the argument matrix ; it does not depend on its eigenvectors. This function provides a measure of the volume of an ellipsoid. Precisely, the volume of the ellipsoid is given by , where is a constant (given by the volume of the unit sphere in ). Thus, . This means that the volume of the ellipsoid is a function of the product of the eigenvalues of the matrix . 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.