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.