Connection between eigenvalues of A and its LDL decomposition

Question

Consider an undirected graph $G$ with $N$ vertices and its adjacency matrix $n_{ij}$: $n_{ij} = 1$ if vertices $i$ and $j$ are connected by an edge and $n_{ij} = 0$ otherwise. Consider $A_{ij} \equiv \left(\sum_{k=1}^N n_{ik}\right)\delta_{ij} - n_{ij}$ and its LDL decomposition \begin{equation} A = L D L^T, \end{equation} where $L$ is lower triangular with unit on the diagonal, and $D$ is diagonal.

Suppose now that $G$ is connected: Then it can be shown that $A$ has only one zero eigenvalue, which we denote by $\lambda_1$, and that there is only one zero diagonal entry in $D$, which we denote by $D_{11}$. By diagonalizing $A$ numerically for multiple $G$s, I find heuristically that the following equation \begin{equation} \prod_{i=2}^N \lambda_i = N \prod_{i=2}^N D_i \end{equation} is numerically satistified.

Is this equation true in general? If it is, do you have any idea of how to prove it?

Thanks

Answer 1

More generally, the following appears to be true. Suppose $A$ is a real $N \times N$ symmetric matrix with rank $N-1$ and $A b = 0$ where $b \ne 0$. Let $A = LDL^T$ be the $LDL^T$ decomposition of $A$ (assuming no pivoting needed), and $d_1, \ldots, d_N$ the diagonal entries of $D$, where $d_N = 0$. If $c_1$ is the coefficient of $\lambda$ in the characteristic polynomial $\det(A - \lambda I)$of $A$, then $$c_1 = d_1 \ldots d_{N-1} \sum_{i=1}^N (b_i/b_N)^2 $$

EDIT: Here is a proof.

Let $L$ be a lower triangular matrix with diagonal entries $1$, $D$ a diagonal matrix with diagonal entries $d_1, \ldots, d_N$ where $d_N = 0$, and $A = L D L^T$. Then $b = (L^T)^{-1} e_N$ is in the null space of $A$, and $\sum_{i=1}^N b_i^2 = b^T b = (L^{-1} (L^T)^{-1} )_{NN} = \left((L^T L)^{-1}\right)_{NN}$. By Cramer's rule that is $\det(C_{NN})$, where $C_{NN}$ is the upper left $(N-1) \times (N-1)$ submatrix of $L^T L$ (note that $\det(L^T L) = (\det L)^2 = 1$). On the other hand, $b_N = 1$ (because $(L^T)^{-1}$ is upper triangular with $1$ on the diagonal).

Now by Sylvester's determinant theorem the characteristic polynomial of $A = L D L^T$ is also the characteristic polynomial of $D L^T L$. Since the last row of $D L^T L$ is $0$, we expand along the last row and find that the coefficient of $\lambda$ in the characteristic polynomial is the determinant of the upper left $(N-1) \times (N-1)$ submatrix of $D L^T L$. This submatrix is the product of the upper left $(N-1) \times (N-1)$ submatrices of $D$ and $L^T L$, so its determinant is the product of the determinants of those, namely $d_1 \ldots d_{N-1} \det(C_{NN})$.