Log-Determinant Function
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See: Symmetric Matrix, Set Function, Entropy Function.
References
2013
- https://inst.eecs.berkeley.edu/~ee127a/book/login/def_logdet_fcn.html
- QUOTE: 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 [math]\displaystyle{ f(X) = left{ begin{array}{ll} log det X & mbox{if } X succ 0, +infty & mbox{otherwise.} end{array} right. \lt P\gt The function can be expressed in terms of the (real, positive) [[eigenvalue]]s of the argument matrix X; it does not depend on its eigenvectors. \lt P\gt This function provides a measure of the volume of an ellipsoid. Precisely, the volume of the ellipsoid: \lt math\gt mathbf{E} = left{ x ~:~ x^TX^{-1}x le 1 right} }[/math] 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.
- QUOTE: 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 [math]\displaystyle{ f(X) = left{ begin{array}{ll} log det X & mbox{if } X succ 0, +infty & mbox{otherwise.} end{array} right. \lt P\gt The function can be expressed in terms of the (real, positive) [[eigenvalue]]s of the argument matrix X; it does not depend on its eigenvectors. \lt P\gt This function provides a measure of the volume of an ellipsoid. Precisely, the volume of the ellipsoid: \lt math\gt mathbf{E} = left{ x ~:~ x^TX^{-1}x le 1 right} }[/math] 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}.