Eigenvalues for block-tridiagonal adjacency matrix with "rays"

Question

I'm looking for closed-form expressions of the eigenvalues of the following adjacency matrix:

\begin{equation} M = \begin{pmatrix} A & B^T & 0\\ B & A & C^T\\ 0 & C & A \end{pmatrix} \end{equation}

where the matrix $A$ is given by

\begin{equation} A = \begin{pmatrix} 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ \end{pmatrix} \end{equation} The matrix $B$ is given by the $9\times9$ matrix with zeros everywhere, except on the first row, which is the vector $v = (0,0,0,0,1,1,0,1,1)$. The matrix $C$ is $C = J B^T J$, where $J$ is the exchange matrix. Hence, one only needs to know $v$ to create $B$ and $C$. Here is a matrix plot of $M$ to show its structure, with the vector $v$ creating "rays".

So now I'm looking for closed-form expressions for the eigenvalues of $M$. I have computed the eigenvalues numerically with Mathematica but I cannot seem to find a recognizable structure in them. The problem is that I want to use this matrix $M$ as the new matrix $A$ for a bigger matrix, with different $v$, in a nested way. So if a closed-form expression can be found involving the spectrum of A (which we can assume to be known), I can then hopefully generate the spectrum iteratively for bigger matrices.

I have tried to find the eigenvalues using the technique from this post in a previous post of mine, but the only answer I got was one I had already found and led me nowhere... A second approach I tried was partitioning the matrix into two arrowhead matrices and one block matrix which was then a submatrix of $A$ with the outer rows/columns removed, but this also yielded no result.

Thanks!

Answer 1

The matrix is symmetric and has non-negative eigenvalues. Therefore, the singular values will be the same as the eigenvalues. You can compute them symbolically with Mathematica.

The characteristic polynomial does not further factor, so I don't think you can get this any simpler (see here).

FullSimplify[CharacteristicPolynomial[matrix, x]] // Factor

-(1 + x)^3 (8 - 35 x - 42 x^2 + 159 x^3 + 66 x^4 - 194 x^5 - 62 x^6 + 
   83 x^7 + 20 x^8 - 15 x^9 - 2 x^10 + x^11) (-8 + 52 x - 15 x^2 - 
   277 x^3 + 135 x^4 + 499 x^5 - 132 x^6 - 404 x^7 + 11 x^8 + 
   133 x^9 + 7 x^10 - 19 x^11 - x^12 + x^13)

$$ \left\{\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,13,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,11,0\right],\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,12,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,10,0\right],\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,11,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,13,0\right],\text{Root}\left[\text{$\#$1}^{11}+2 \text{$\#$1}^{10}-15 \text{$\#$1}^9-20 \text{$\#$1}^8+83 \text{$\#$1}^7+62 \text{$\#$1}^6-194 \text{$\#$1}^5-66 \text{$\#$1}^4+159 \text{$\#$1}^3+42 \text{$\#$1}^2-35 \text{$\#$1}-8\&,11,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,9,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,12,0\right],\text{Root}\left[\text{$\#$1}^{11}+2 \text{$\#$1}^{10}-15 \text{$\#$1}^9-20 \text{$\#$1}^8+83 \text{$\#$1}^7+62 \text{$\#$1}^6-194 \text{$\#$1}^5-66 \text{$\#$1}^4+159 \text{$\#$1}^3+42 \text{$\#$1}^2-35 \text{$\#$1}-8\&,10,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,11,0\right],\text{Root}\left[\text{$\#$1}^{11}+2 \text{$\#$1}^{10}-15 \text{$\#$1}^9-20 \text{$\#$1}^8+83 \text{$\#$1}^7+62 \text{$\#$1}^6-194 \text{$\#$1}^5-66 \text{$\#$1}^4+159 \text{$\#$1}^3+42 \text{$\#$1}^2-35 \text{$\#$1}-8\&,9,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,10,0\right],\text{Root}\left[\text{$\#$1}^{11}+2 \text{$\#$1}^{10}-15 \text{$\#$1}^9-20 \text{$\#$1}^8+83 \text{$\#$1}^7+62 \text{$\#$1}^6-194 \text{$\#$1}^5-66 \text{$\#$1}^4+159 \text{$\#$1}^3+42 \text{$\#$1}^2-35 \text{$\#$1}-8\&,8,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,9,0\right],1,1,1,\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,10,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,8,0\right],\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,9,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,7,0\right],\text{Root}\left[\text{$\#$1}^{13}+\text{$\#$1}^{12}-19 \text{$\#$1}^{11}-7 \text{$\#$1}^{10}+133 \text{$\#$1}^9-11 \text{$\#$1}^8-404 \text{$\#$1}^7+132 \text{$\#$1}^6+499 \text{$\#$1}^5-135 \text{$\#$1}^4-277 \text{$\#$1}^3+15 \text{$\#$1}^2+52 \text{$\#$1}+8\&,8,0\right],\text{Root}\left[\text{$\#$1}^{11}+2 \text{$\#$1}^{10}-15 \text{$\#$1}^9-20 \text{$\#$1}^8+83 \text{$\#$1}^7+62 \text{$\#$1}^6-194 \text{$\#$1}^5-66 \text{$\#$1}^4+159 \text{$\#$1}^3+42 \text{$\#$1}^2-35 \text{$\#$1}-8\&,7,0\right],\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,8,0\right],\text{Root}\left[\text{$\#$1}^{11}-2 \text{$\#$1}^{10}-15 \text{$\#$1}^9+20 \text{$\#$1}^8+83 \text{$\#$1}^7-62 \text{$\#$1}^6-194 \text{$\#$1}^5+66 \text{$\#$1}^4+159 \text{$\#$1}^3-42 \text{$\#$1}^2-35 \text{$\#$1}+8\&,6,0\right],\text{Root}\left[\text{$\#$1}^{13}-\text{$\#$1}^{12}-19 \text{$\#$1}^{11}+7 \text{$\#$1}^{10}+133 \text{$\#$1}^9+11 \text{$\#$1}^8-404 \text{$\#$1}^7-132 \text{$\#$1}^6+499 \text{$\#$1}^5+135 \text{$\#$1}^4-277 \text{$\#$1}^3-15 \text{$\#$1}^2+52 \text{$\#$1}-8\&,7,0\right]\right\} $$