Stahl's theorem

Let ${\displaystyle \operatorname {tr} }$ denote the trace of a matrix. If ${\displaystyle A}$ and ${\displaystyle B}$ are ${\displaystyle n\times n}$ Hermitian matrices and ${\displaystyle B}$ is positive semidefinite, define ${\displaystyle \mathbf {f} (t)=\operatorname {tr} (\exp(A-tB))}$, for all real ${\displaystyle t\geq 0}$. Then ${\displaystyle \mathbf {f} }$ can be represented as the Laplace transform of a non-negative Borel measure ${\displaystyle \mu }$ on ${\displaystyle [0,\infty )}$. In other words, for all real ${\displaystyle t\geq 0}$,

${\displaystyle \mathbf {f} }$(t) = ${\displaystyle \int _{[0,\infty )}e^{-ts}\,d\mu (s)}$,

for some non-negative measure ${\displaystyle \mu }$ depending upon ${\displaystyle A}$ and ${\displaystyle B}$.^[6]