I am going to construct a matrix of the following form $$ T_n = \begin{bmatrix} a_1 & b_1 & 0 & 0 & 0 & 0\\ b_1 & a_2 & b_2 & 0 & 0 & 0 \\ 0& b_2 & a_2 & b_3 & 0 & 0 \\ 0 & 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & 0 & b_{n-2} & a_{n-1} & b_{n-1} \\ 0 & 0 & 0 & \cdots & b_{n-1} & a_n \end{bmatrix} $$
from the following eigen information:
- One eigenvalue of each of the leading principal's minors of $T_n$, named $\lambda^{(i)}$ for the $i$th leading principal minor
- One eigenpair of $T_n$, named eigenvector $X$ and eigenvalue $\lambda^{(n)}$
The solution to this problem is simple and is given in this link. However, I have the restriction that I cannot use $T_nX = \lambda^{(n)}X$ for the first row.
My Solution
The roots of $P_{n}(\lambda) = \det(\lambda I_n-T_n)$ are the eigenvalues of $T_n$ and \begin{equation} P_{j}(\lambda) = (\lambda - a_j)P_{j-1}(\lambda) - b_{j-1}^2P_{j-2}(\lambda), j=3, 4, \cdots, n \end{equation}
Therefore the solution is the following system of equations
\begin{align} P_{j}(\lambda^{(j)}) &= 0 \\ T_nX &= \lambda^{(n)}X \end{align}
where $P_j(\lambda) = \det(\lambda I_j - T_j)$ and $T_j$ is the $j$th leading principal minor of $T_n$, i.e.
$$ T_j = \begin{bmatrix} a_1 & b_1 & 0 & 0 & 0 & 0\\ b_1 & a_2 & b_2 & 0 & 0 & 0 \\ 0& b_2 & a_2 & b_3 & 0 & 0 \\ 0 & 0 & \ddots & \ddots & \ddots & 0 \\ 0 & 0 & 0 & b_{j-2} & a_{j-1} & b_{j-1} \\ 0 & 0 & 0 & \cdots & b_{j-1} & a_j \end{bmatrix} $$
From $P_{1}(\lambda^{(1)}) = 0$, it is easy to see $a_1 = \lambda^{(1)}$. For the second row: \begin{align} (\lambda^{(n)} - a_2)(\lambda^{(n)} - a_1) &= b_1^2\\ b_1x_1 + b_2x_3 &= (\lambda^{(n)} - a_2)x_2 \end{align}
and it gives
\begin{equation} b_1^2\dfrac{x_2}{(\lambda^{(n)} - \lambda^{(1)})} - b_1x_1 - b_2x_3 = 0 \end{equation}
Unfortunately, I can't solve this because $b_2$ is also an unknown variable. If we continue this way, we will end up with
\begin{align} (\lambda^{(n)} - a_n)P_{n-1}(\lambda^{(n)}) &= b_{n-1}^2P_{n-2}(\lambda^{(n)})\\ b_{n-1}x_{n-1} &= (\lambda^{(n)} - a_n)x_n \end{align}
But I couldn't find a solution yet. I would like to know if it is possible to find a solution to this problem with the given data. Are the input enough to solve the problem? and if yes, what is the answer?
Thank you so much for your attention and participation.