The matrix, analytically constructed based on the hypothese, that the eigenvalues are the consecutive powers of u = log(t)
\( \hspace{24}
\begin{matrix} {rrr}
1.00000000000 & 1.00000000000 & 1.00000000000 \\
0 & 1.57079632679*I & 3.14159265359*I \\
0 & -1.23370055014 & -4.93480220054 \\
0 & -0.645964097506*I & -5.16771278005*I \\
0 & 0.253669507901 & 4.05871212642 \\
0 & 0.0796926262462*I & 2.55016403988*I \\
0 & -0.0208634807634 & -1.33526276886-2.13383011501E-12*I \\
0 & -0.00468175413532*I & 9.74997503272E-12-0.599264529316*I \\
0 & 0.000919260274839 & 0.235330630320+1.03824464497E-11*I \\
0 & 0.000160441184787*I & 7.34062412572E-11+0.0821458865076*I \\
0 & -0.0000252020423730 & -0.0258068913636+0.000000000336008212438*I \\
0 & -0.00000359884323522*I & -0.000000000531166859357-0.00737043148889*I \\
0 & 0.000000471087478009 & 0.00192957576096+1.44905471842E-10*I \\
0 & 0.0000000569217294451*I & -0.00000000197293927481+0.000466304152891*I \\
0 & -0.00000000638660319694 & -0.000104637211800-0.00000000321239246183*I \\
0 & -0.000000000668803312296*I & 0.00000000165543902983-0.0000219117619420*I \\
0 & 6.56592442263E-11 & 0.00000429946860414-0.00000000166180488357*I \\
0 & 6.06668588155E-12*I & 0.00000000320792169496+0.000000794286619384*I \\
0 & -1.00000000000E-12 & -0.000000140053803989+0.00000000195676397990*I \\
0 & 0 & -2.05225455659E-10-0.0000000242612901612*I \\
0 & 0 & 0.00000000407962123588+0.000000000376346155829*I \\
0 & 0 & -2.13051400732E-10+0.000000000561705712732*I \\
0 & 0 & -4.37085349142E-11-4.53336927688E-11*I \\
0 & 0 & 3.32471067492E-12-1.00000000000E-12*I
\end{matrix} \)
(the second column are the interesting ones, they serve as coefficients for the powerseries in x for the expression {I,x}^^I ), and if x=1 they simply must be summed )
And the partial-sums of the second column should converge to y=I, since this means y = {I,1}^^I, :
\( \hspace{24}
\begin{matrix} {rrr}
0.977995110024+0.146699266504*I \\
0.570308500749+1.47507813506*I \\
-0.114402540375+1.14713275303*I \\
-0.0106044591818+0.993956473385*I \\
-0.00229741633360+1.00160960663*I \\
-0.000143765522959+0.999877073257*I \\
-0.0000575116782223+1.00002476990*I \\
-0.000000580602181001+1.00000205224*I \\
-0.00000100453146370+1.00000024808*I \\
-0.00000000833987995204+1.00000016631*I \\
0.00000000258190521640+1.00000000353*I \\
-0.000000000512676097081+1.00000000285*I \\
0.000000000486292953022+1.00000000023*I \\
2.59309575160E-11+0.999999999974*I \\
4.18678540668E-12+1.00000000000*I \\
1.17217031429E-12+0.999999999999*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I
\end{matrix} \)
This is also good for integer powers of this matrix; but again, for fractional or even complex powers the obtainable results for this parameters with that number of terms (24(last document) or 32) are not convincing.
Gottfried
\( \hspace{24}
\begin{matrix} {rrr}
1.00000000000 & 1.00000000000 & 1.00000000000 \\
0 & 1.57079632679*I & 3.14159265359*I \\
0 & -1.23370055014 & -4.93480220054 \\
0 & -0.645964097506*I & -5.16771278005*I \\
0 & 0.253669507901 & 4.05871212642 \\
0 & 0.0796926262462*I & 2.55016403988*I \\
0 & -0.0208634807634 & -1.33526276886-2.13383011501E-12*I \\
0 & -0.00468175413532*I & 9.74997503272E-12-0.599264529316*I \\
0 & 0.000919260274839 & 0.235330630320+1.03824464497E-11*I \\
0 & 0.000160441184787*I & 7.34062412572E-11+0.0821458865076*I \\
0 & -0.0000252020423730 & -0.0258068913636+0.000000000336008212438*I \\
0 & -0.00000359884323522*I & -0.000000000531166859357-0.00737043148889*I \\
0 & 0.000000471087478009 & 0.00192957576096+1.44905471842E-10*I \\
0 & 0.0000000569217294451*I & -0.00000000197293927481+0.000466304152891*I \\
0 & -0.00000000638660319694 & -0.000104637211800-0.00000000321239246183*I \\
0 & -0.000000000668803312296*I & 0.00000000165543902983-0.0000219117619420*I \\
0 & 6.56592442263E-11 & 0.00000429946860414-0.00000000166180488357*I \\
0 & 6.06668588155E-12*I & 0.00000000320792169496+0.000000794286619384*I \\
0 & -1.00000000000E-12 & -0.000000140053803989+0.00000000195676397990*I \\
0 & 0 & -2.05225455659E-10-0.0000000242612901612*I \\
0 & 0 & 0.00000000407962123588+0.000000000376346155829*I \\
0 & 0 & -2.13051400732E-10+0.000000000561705712732*I \\
0 & 0 & -4.37085349142E-11-4.53336927688E-11*I \\
0 & 0 & 3.32471067492E-12-1.00000000000E-12*I
\end{matrix} \)
(the second column are the interesting ones, they serve as coefficients for the powerseries in x for the expression {I,x}^^I ), and if x=1 they simply must be summed )
And the partial-sums of the second column should converge to y=I, since this means y = {I,1}^^I, :
\( \hspace{24}
\begin{matrix} {rrr}
0.977995110024+0.146699266504*I \\
0.570308500749+1.47507813506*I \\
-0.114402540375+1.14713275303*I \\
-0.0106044591818+0.993956473385*I \\
-0.00229741633360+1.00160960663*I \\
-0.000143765522959+0.999877073257*I \\
-0.0000575116782223+1.00002476990*I \\
-0.000000580602181001+1.00000205224*I \\
-0.00000100453146370+1.00000024808*I \\
-0.00000000833987995204+1.00000016631*I \\
0.00000000258190521640+1.00000000353*I \\
-0.000000000512676097081+1.00000000285*I \\
0.000000000486292953022+1.00000000023*I \\
2.59309575160E-11+0.999999999974*I \\
4.18678540668E-12+1.00000000000*I \\
1.17217031429E-12+0.999999999999*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I \\
1.00000000000*I
\end{matrix} \)
This is also good for integer powers of this matrix; but again, for fractional or even complex powers the obtainable results for this parameters with that number of terms (24(last document) or 32) are not convincing.
Gottfried
Gottfried Helms, Kassel