\(e\) to the \(A\), in a new way (Q5954540)

Let \(A\) be a complex traceless hermitian matrix of size \(d \times d\) and \(W\) a hermitian projector of rank 1 in \(C^d\), \(W_{ij}=v_i {\overline v}_j\), where \(v_i\) is a unit vector in \(C^d\). Denote by \(\int d\Omega\) the normalized integral over all such \(W\) defined by integrating the associated \(v_i\) over a unit sphere in \(C^2\) with the normalized unitary-invariant measure. For the case \(d=2\) the author presents the formula NEWLINE\[NEWLINEe^A=\int d\Omega e^{\text{Tr}(AW)} [I+4(W-1/2 I)+\text{Tr}(AW) I+ 2\text{Tr}(AW)(W-1/2 I)].NEWLINE\]NEWLINE Similar expressions are obtained for \(d=3, 4\).