I have a matrix integration problem. It is based on the first integral under the section, "energy transfer efficiency and transport time" in the article, environment-assisted transport. There is a function, $\rho(t)$ that is the most important in calculating the integral but is a time-dependent matrix. Now, I have the function evaluated at various times. With that, I want to calculate the integral presented in the article. I don't want to find the time-dependent matrix but use the time-dependent matrix evaluated at various times to approximate the integral. Is this possible?
Here is the integral:
$$2\sum_m\kappa_m\int^\infty_0\langle m|\rho(t)|m \rangle dt$$ where $\kappa$ is a constant.
EDIT:
Let me reword my question. Let's say we have three matrices, $\rho(t)$, a 7x7 matrix, and 7x1 and 1x7 matrices ($|m\rangle$ and $\langle m|$). Now, I want to take the product of all these matrices and take the integral of it from zero to infinity. However, even though $\rho(t)$ is a time-dependent matrix, I only have it evaluated at particular periods of time, not as an expression in each element. Now how can I do this integral without regressing each element of the matrix?