This is not an answer but an explanation to my comment above
I begin with an integer 3x3-matrix M. Example $ \qquad \small M= \begin{array} {rrr} 75 & 46 & 170 \\ 113 & 193 & 43 \\ 23 & 38 & 90 \end{array} $
Then I get the rotationmatrix $T$ which rotates $M$ to lower triangular shape by column-rotation (M T = lower triangular) :
$ \qquad \small T = \begin{array} {rrr} 0.391811884067 & 0.332906301267 & -0.857704402508 \\ 0.240311288894 & 0.862849514418 & 0.444681009149 \\ 0.888106937218 & -0.380347354460 & 0.258073551572 \end{array} $
The mercatorseries for the log of $T$ converges sufficiently fast, I get
$ \qquad \small L= \begin{array} {rr} 0 & 0.0628202506390 & -1.18442995060 \\ -0.0628202506390 & 0 & 0.559733048879 \\ 1.18442995060 & -0.559733048879 & 0 \end{array}$
using 200 terms and all displayed digits correct.
Then the $0.2$-step of that rotation is $T02=\exp(0.2*L) $
$ \qquad \small T02 = \begin{array} {rrr} 0.972024543477 & 0.0256039075816 & -0.233479606807 \\ 0.000762973571737 & 0.993691347552 & 0.112146884361 \\ 0.234878063577 & -0.109187662843 & 0.965872843356 \end{array}$
Using the mateigen-procedure in Pari/GP I get complex-valued matrices of eigenvectors and the following complex-valued diagonal-matrix $D$:
\qquad \small D= \begin{array} {rrr} 1.00000000000 & . & . \\ . & 0.256367475028-0.966579390297 î & . \\ . & . & 0.256367475028+0.966579390297 î \end{array}
(which finally provides the same result $T02$, if I ask Pari/GP for the $0.2$'th-power of the scalar diagonal-entries in $D$, however with spurious imaginary entries.)
The sequence of 5 steps of the rotation ($0,0.2,0.4,0.6,0.8,1.0$) are then
$ \qquad \small \begin{array} {rrr} 75 & 46 & 170 \\ 113 & 193 & 43 \\ 23 & 38 & 90 \\ - & - & - \\ 112.866208353 & 29.0681923728 & 151.846169541 \\ 120.085784046 & 189.980602132 & 36.7936853768 \\ 43.5245832176 & 28.5222714255 & 85.8201065512 \\ - & - & - \\ 145.396237174 & 15.1948988538 & 123.572040788 \\ 125.513309167 & 187.839329344 & 28.8061724883 \\ 62.4859853141 & 20.0862368751 & 75.9278916310 \\ - & - & - \\ 170.364666021 & 5.32920901101 & 87.1118826728 \\ 128.911271501 & 186.722668818 & 19.5838972005 \\ 78.5870327743 & 13.2690161490 & 61.0000941817 \\ - & - & - \\ 186.063373076 & 0.146047172654 & 44.9599807576 \\ 130.047212143 & 186.707012713 & 9.75776703751 \\ 90.7262325637 & 8.53698394520 & 42.0579437196 \\ - & - & - \\ 191.418389921 & 0 & 0 \\ 128.843419956 & 187.793432084 & 0 \\ 98.0731266611 & 6.21386457561 & 20.3972967315 \end{array} $