# Questions tagged [matrix-exponential]

"The matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function."

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### On the weighted adjacency matrix of a directed acyclic graph (DAG)

A matrix $W \in \mathbb{R}^{d \times d}$ is the weighted adjacency matrix of a directed acyclic graph (DAG) if and only if $$h(W) = \operatorname{tr} \left( \exp(W \circ W ) \right) - d = 0$$ where ...
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### Positivity of semigroup $(P_t)_{t\geq0}$ implies its contractivity, $\| P_tf\| \leq \|f\|$

Let $\Omega$ be a Polish space, denote $E = (C_b(\Omega),\|\cdot\|)$ be the Banach space of continuous and bounded functions with the usual supremum norm. A family $(P_t)_{t\geq 0}$ of linear ...
Suppose I have $n$ operators $X_1,\ldots,X_n$, which are closed under commutators: $$[X_i,X_j]=c_1X_1+\cdots+c_nX_n,$$ where $c_j$ are constants (which will depend on $i$ and $j$). Now consider a ...