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Consider a real, symmetric matrix $M(t)$ with time-varying elements. Each of the elements are zero for $t<0$ and positive for $t\geq 0$. Assume that $M(t)\succeq 0 \;\forall t\geq 0$ (i.e., it is positive semidefinite).

Is the following matrix obtained by taking Laplace transform of $M(t)$ also positive semidefinite? $$ \int_{t=0}^{\infty}e^{-s t} M(t) dt. $$ Here $s$ is a complex number.

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  • $\begingroup$ positive semidefinite for all values of $s$? Including zero? If positive definite the integral would blow up for real negative $s$? $\endgroup$ – percusse Jun 7 '18 at 21:33
  • $\begingroup$ No, only for those values of $s$ for which the Laplace transform makes sense. $\endgroup$ – K. Ghusinga Jun 8 '18 at 3:23

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