There is a $N \times N$ diagonal matrix A with an infinite cycle with a period of 3, and an infinite cyclic matrix B, as follows. Assume $\Phi$ is a matrix of eigenvectors of $A^{-1}B$, where $\Phi^{T}A\Phi=I$, I is the $N \times N$ identity matrix.
Currently, a $N \times N$ diagonal matrix C has been constructed as follows, the elements of A is redistributed while keeping the sum of the element same. How can I get the general rule of the diagnonal element of matrix $\Phi^{T}C\Phi$?
Through numerical calculations, I have found that when N is is sufficiently large, the first diagonal element of $\Phi^{T}C\Phi$ is 1. How can this be proven?
A=\begin{bmatrix} a_1 & 0 & 0 & \cdots & & & \\ 0 & a_2 & 0 & \cdots & & & \\ 0 & 0 & a_3 & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \\ & & & & a_1 & 0 & 0 \\ & & & & 0 & a_2 & 0 \\ & & & & 0 & 0 & a_3 \\ & & & & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}
B=\begin{bmatrix} 2b & -b & 0 & \cdots & & & \\ -b & 2b & -b & \cdots & & & \\ 0 & -b & 2b & -b & \cdots & & \\ \vdots & \vdots & -b & 2b & -b & \cdots & \\ & & \vdots & -b & 2b & -b & 0 \\ & & & \vdots & -b & 2b & -b \\ & & & & 0 & -b & 2b & \ddots \\ & & & & \vdots & \vdots & \ddots & \ddots \\ \end{bmatrix}
C=\begin{bmatrix} 0 & 0 & 0 & \cdots & & & \\ 0 & 0 & 0 & \cdots & & & \\ 0 & 0 & a_1+a_2+a_3 & \cdots & & & \\ \vdots & \vdots & \vdots & \ddots & & & \\ & & & & 0 & 0 & 0 \\ & & & & 0 & 0 & 0 \\ & & & & 0 & 0 & a_1+a_2+a_3 \\ & & & & \vdots & \vdots & \vdots & \ddots \\ \end{bmatrix}