Jacobi行列式的计算

随机矩阵之间变换的Jacobi行列式的计算,常规方法就是求出变换的行列式的元素再求行列式值,这一方法能计算许多变换的Jacobi,但其计算量非常大,有时甚至无法算出结果.该文充分利用外微分形式的特殊性质,巧妙地计算了几个重要的Jacobi行列式,并且给出了众多文献都引用了但都没有给出证明的变换Y=X'DX的Jacobi行列式J(X→Y)=2-m|D|-1/2|Y|-1/2∏(li+lj)-1.利用Muirhead提出的外微分方法计算了几个重要的Jacobi行列式.

Some Jacobians of the Transformation between Random Vectors and Matrices

In distribution theory,functions of random vectors and matrices will be of interest and we will need to know how density functions are transformed. This involves computing the Jacobians of these transformations. We know the Jacobian of the transformation from X to Y, is .Often when dealing with many variables it is tedious to explic- ityly write out the determinant det ( ) . This paper use the exterior product and exterior differential forms to treat the proceeding problem. Caculates some Jacobians of particular interest to us. The main work is the proof of Theorem 6.