I have matrices like A and B. Rows $2-(n-1)$ are stochastic, row $n$ is sub stochastic and the first row differs always in the first entry.
I am interested in the largest eigenvalue ($\lambda_1$) of these matrices.
In case the first entry is $\frac{1}{2}$ (matrix A): $\lambda_1(A) = \cos (\frac{\pi}{2n+1})$.
For matrix B, $\lambda_1(B) = 1$.
$A = \begin{bmatrix} \frac{1}{2}& \frac{1}{2} & 0 & 0 & \dots & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & \dots & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & 0 \\ 0 & 0 & 0 &0 & \frac{1}{2} & 0 \end{bmatrix}$
$B = \begin{bmatrix} \frac{1}{2-\frac{2}{n}}& \frac{1}{2} & 0 & 0 & \dots & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & \dots & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & 0 \\ 0 & 0 & 0 &0 & \frac{1}{2} & 0 \end{bmatrix}$
Can I use this knowledge to get a good estimate on the eigenvalue of a matrix C in which the first entry is between $\frac{1}{2}$ and $\frac{1}{2-\frac{2}{n}}$?
$C = \begin{bmatrix} \frac{1}{2-\frac{2}{n^2}}& \frac{1}{2} & 0 & 0 & \dots & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & \dots & 0 \\ 0 & \frac{1}{2} & 0 & \frac{1}{2} & \dots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & 0 \\ 0 & 0 & 0 &0 & \frac{1}{2} & 0 \end{bmatrix}$