I am trying to apply some methods in a paper and I have to solve the following fixed point equation from Proposition VIII.4.3 in Asmussen (2000):
$$\mu_+ =\mu \int_0^{\infty}exp((\mathbf{S}+\mathbf{s}\mu_+)y)A(dy))= \mu \hat{A}(\mathbf{S}+\mathbf{s}\mu_+) $$
We work with: $$\mu_+ = \mu \hat{A}(\mathbf{S}+\mathbf{s}\mu_+) $$
It is given that $\mu_+ $and $\mu$ are $1 \times 2$ vectors, $\mathbf{s}$ is a $2 \times 1 $ vector, S is a $2\times2$ matrix and I have calculated that $$\hat{A}(z)=\frac{5.5}{5.5+4z-4.5z^2}\times \frac{p+z}{p}$$ where $p$ is a constant. So obviously $\mathbf{S}+\mathbf{s}\mu_+$ is a $2\times 2$ matrix. Let $V=\mathbf{S}+\mathbf{s}\mu_+$
My question is how do I apply the function $\hat{A}(\cdot)$ onto $V$. Do I apply the function $\hat{A}(\cdot)$ on each element of $V$ individually to get the resultant matrix?
For e.g. if $$V= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$
Then is $$ \hat{A}(V)= \begin{bmatrix} \hat{A}(1) & \hat{A}(2) \\ \hat{A}(3) & \hat{A}(4) \end{bmatrix} $$
Or is this wrong and is there a correct way of applying the function here?