Define the Laplacian matrix as $L = D – A$. Here, $A$ is the adjacency matrix of a directed weighted graph with $n$ vertices so that the entries $A_{ij}$ of $A$ are equal to a positive weight if there is an arrow form the vertex $j$ to $i$ and $0$ otherwise, and $D = \operatorname{diag}(\sum_{i=1}^n A_{i1},\cdots,\sum_{i=1}^n A_{in})$.
Can one come up with an example of weighted directed graph for which $L$ would be defective matrix?
Here, by "defective matrix" I mean a square matrix that does not have a complete basis of eigenvectors, and is therefore not diagonalizable.
Best Answer
The graph $1 \to 2 \to \dots \to n$ has Laplacian matrix $$ L = \pmatrix{ 1&\\ -1&1&\\ &-1&\ddots\\ && \ddots&1\\ &&&-1&0 } $$ where the unwritten entries are zeros. This matrix is defective for $n \geq 3$.