derivative of matrix
Suppose is an open set of , and for each , isan matrix. If each element in is a differentiable functionof , we say that is a differentiable, and define thederivative of componentwise. This derivative we shall write as or .
Properties
In the below we assume that all matrices are dependent on a parameter and the matrices are differentiable with respect to .
- 1.
For any matrix ,
where is the matrix transpose.
- 2.
If are matrices such that is defined, then
- 3.
When is invertible
,
- 4.
For a square matrix
,
where is the matrix trace.
- 5.
If are matrices and is theHadamard product of and , then