If $A=(a_{ij})$ is an $n\times n$ symmetric positive matrix, is it possible to calculate the following matrix integral?
$$\int_{0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |\mathrm dt,$$
where $\left |\cdot \right |$ denotes the absolute operator of the matrix like abs() function in matlab?
Would the problem become easier if $t\in \mathbb{Z}$? In this case, the matrix integral could be rewritten as:
$$\sum_{t=0}^{\infty}\left | e^{-A(t+1))}-e^{-At)} \right |?$$
Many thanks for your help!!!
The abs function in Matlab operates entry-by-entry on a matrix, producing another matrix of the same shape. Do I understand you correctly, then, that you want the integrals of the absolute value of each entry of $e^{-A(t+1)} - e^{-At}$?
In general I don't think that's going to be pleasant. However, if you can figure out the intervals of $t$ on which the entry is positive or negative, you can calculate it: for example if it's positive for $a < t < b$, $$ \int_a^b \left(e^{-A(t+1)} - e^{-At}\right) \ dt = (e^{-A}-I) \int_a^b e^{-At}\ dt = (e^{-A} - I) A^{-1} (e^{-Ab} - e^{-Aa})$$