I am trying to analytically find the inverse of a matrix given by: \begin{align} W = \left( I - \alpha e^A \right)^{-1}, \end{align} where $I$ is the identity matrix of appropriate size, $e^A$ denotes elementwise exponential of $A$ and \begin{align} A = \begin{bmatrix} \frac{ (1-1)^2 }{ \sigma^2 } & \frac{ (2-1)^2 }{ \sigma^2 } & \frac{ (3-1)^2 }{ \sigma^2 } & \dots & \frac{ (n-1)^2 }{ \sigma^2 } \\ \frac{ (1-2)^2 }{ \sigma^2 } & \frac{ (2-2)^2 }{ \sigma^2 } & \frac{ (3-2)^2 }{ \sigma^2 } & \dots & \frac{ (n-2)^2 }{ \sigma^2 } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \frac{ (1-N)^2 }{ \sigma^2 } & \frac{ (2-N)^2 }{ \sigma^2 } & \frac{ (3-N)^2 }{ \sigma^2 } & \dots & \frac{ (N-N)^2 }{ \sigma^2 } \\ \end{bmatrix}, \end{align} so that I'm trying to find \begin{align} W = \begin{bmatrix} 1 - \alpha \exp{ \left( \frac{ 0^2 }{ \sigma^2 } \right) } & - \alpha \exp{ \left( \frac{ 1^2 }{ \sigma^2 } \right) } & - \alpha \exp{ \left( \frac{ 2^2 }{ \sigma^2 } \right) } & \dots & - \alpha \exp{ \left( \frac{ (n-1)^2 }{ \sigma^2 } \right) } \\ - \alpha \exp{ \left( \frac{ 1^2 }{ \sigma^2 } \right) } & 1 - \alpha \exp{ \left( \frac{ 0^2 }{ \sigma^2 } \right) } & - \alpha \exp{ \left( \frac{ 1^2 }{ \sigma^2 } \right) } & \dots & - \alpha \exp{ \left( \frac{ (n-2)^2 }{ \sigma^2 } \right) } \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ - \alpha \exp{ \left( \frac{ (1-N)^2 }{ \sigma^2 } \right) } & - \alpha \exp{ \left( \frac{ (2-N)^2 }{ \sigma^2 } \right) } & - \alpha \exp{ \left( \frac{ (3-N)^2 }{ \sigma^2 } \right) } & \dots & 1 - \alpha \exp{ \left( \frac{ (N-N)^2 }{ \sigma^2 } \right) } \\ \end{bmatrix}^{-1}. \end{align}
Any help would be much appreciated!
Thank you very much,
Katie