## Wednesday, 5 January 2022

### THE JACOBIAN OF THE MATRIX EXPONENTIAL TRANSFORMATION: BAYESIAN INFERENCE FOR A COVARIANCE MATRIX

Senior co-author  Jan Magnus

Let       C= exp (A)

where C is a positive definite covariance matrix and A=log C denotes the matrix logarithm of C

Then

were the first to complete the monumental task of deriving a closed form expression for the Jacobian of this transformation. My congratulations to the authors!!

For example,

(1) If C possesses an inverted Wishart distribution, then the density of A may be stated in closed form.

(2) If the upper triangular elements of A possess a matrix normal distribution, then the density of C cam be stated in closed form.

This result will find many applications in Bayesian Inference for a Covariance Matrix. The large previous literature includes: