I have a tridiagonal matrix with negative diagonal and and non-negative off-diagonal elements where every row sums to zero (it is the generator to an M/M/n/n queue). My experiments indicate that $|A|^p $ = $|A^p|$, where $|\cdot|$ is the absolute value of every element, but I am not able to prove this.
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$\begingroup$ @user8675309 How do you get $\exp(A)=\exp|A|$ ? Do you think it follows from $|A^p|=|A|^p$ ? It is not true for numbers. $\endgroup$– Ryszard SzwarcMay 22 at 18:03
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$\begingroup$ @RyszardSzwarc I am not sure I get your point. If $|A^p| = |A|^p$ then I guess that $exp(A) = exp|A|$. $\endgroup$– Andreas DahlbergMay 22 at 18:59
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$\begingroup$ @AndreasDahlberg $e^{-1}\neq e^{|-1|}.$ $\endgroup$– Ryszard SzwarcMay 22 at 19:03
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$\begingroup$ @RyszardSzwarc Yes, generally it is not true but I am interested in the case when $A$ is the generator to an M/M/n/n queue. $\endgroup$– Andreas DahlbergMay 22 at 19:15
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Let $D$ denote the diagonal matrix with entries $(-1)^i$ on the main diagonal, i.e. $$d_{ij}=\begin{cases}(-1)^i & j=i\\ 0 & j\neq i \end{cases} $$ Observe that the matrix $DAD$ has nonpositive entries and $|DAD|=|A|.$ Therefore $|A|=-DAD.$ We have $D^2=I.$ Hence $(DAD)^k=DA^kD.$ Therefore $|(DAD)^k|=|DA^kD|=|A^k|.$ Thus $$|A|^k=(-DAD)^k=(-1)^kDA^kD=|DA^kD|=|A^k|$$
Remark The assumption that the rows sum up to $0$ is not necessary for the conclusion.
answered May 22 at 19:22