I am currently investigating the eigenvalue computation of a matrix with a special structure. Consider an $N$-dimensional matrix where all off-diagonal elements are $a$, diagonal elements are typically $b$, but there is a specific diagonal position $k,k$ where the element is $c$. The matrix $A$ can be described as follows for a $N \times N$ matrix with elements $A_{ij}$ defined by:
- If $i = j$ and $i \neq k$, then $A_{ij} = b$
- If $i = j$ and $i = k$, then $A_{ij} = c$
- If $i \neq j$, then $A_{ij} = a$
The general form of the matrix is:$$\begin{bmatrix}b & a & \cdots & a & \cdots & a \\a & b & \cdots & a & \cdots & a \\\vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\a & a & \cdots & c & \cdots & a \\\vdots & \vdots & \ddots & \vdots & \ddots & \vdots \\a & a & \cdots & a & \cdots & b\end{bmatrix}$$
For instance, in a $4 \times 4$ matrix where $a=1$, $b=2$, $c=3$, and $c$ is positioned at the third row and column ($k=3$), the matrix takes the following specific form:$$\begin{bmatrix}2 & 1 & 1 & 1 \\1 & 2 & 1 & 1 \\1 & 1 & 3 & 1 \\1 & 1 & 1 & 2 \\\end{bmatrix}$$The eigenvalues for this matrix are:
- $\lambda_1 = 5.3028$
- $\lambda_2 = 1.6972$
- $\lambda_3,\lambda_4 = 1$
It is known that if the values of $b$ and $c$ are the same, then the matrix would have eigenvalues:
- $\lambda_1 = b + (N-1)a$
- $\lambda_2, \lambda_3, ..., \lambda_N = b-a$
However, in cases where $ a \neq b \neq c$, how can we compute the eigenvalues for such matrices? I would appreciate any methods or references you could provide.