Call a matrix anti-Jordan if it has $-1$s on the subdiagonal, values on the diagonal, and zeroes elsewhere; that is, it is written
$$\begin{bmatrix}
\lambda_1 & 0 & 0 & \dots \\
-1 & \lambda_2 & 0 & \dots\\
0 & -1 & \lambda_3 & \dots\\
\vdots &&& \ddots
\end{bmatrix}$$
Is there a general formula for the inverse of such a matrix?
If it helps, I'm trying to compute the spectral radius of an $N \times N$ matrix $FV^{-1}$ where
$$F = \begin{bmatrix}\beta_1 & \beta_2 & \beta_3 & \dots\\0 & 0 & 0 &\dots\\0 &0 & 0 & \dots\\\vdots &&& \ddots\end{bmatrix}$$
$N$ is "very large" and $$V = \begin{bmatrix}
1 + \mu_1 & 0 & 0 & \dots \\
-1 & 1 + \mu_2 & 0 & \dots\\
0 & -1 & 1 + \mu_3 & \dots\\
\vdots &&& \ddots
\end{bmatrix}.$$
Best Answer
The inverse is the matrix $B = A^{-1}$ with $b_{ij}=0$ for $i < j$ and $b_{ij} = \lambda_j^{-1}\dots\lambda_i^{-1}$ for $i\ge j$.
Your spectral radius will be the absolute value of $$ \sum_{i=1}^N\frac{\beta_i}{(1+\mu_1)\dots(1+\mu_i)}. $$