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→Definition: Introduce "log B" notation. |
→The logarithm of a non-diagonalizable matrix: Expand example to show that the given matrix is actually the logarithm. |
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This [[series (mathematics)|series]] has a finite number of terms (''K''<sup>''m''</sup> is zero if ''m'' is equal to or greater than the dimension of ''K''), and so its sum is well-defined.
'''Example.''' Using this approach, one finds
:<math>\log \begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}
=\begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix}
which can be verified by plugging the righ-hand side into the matrix exponential:
<math display="block">
\exp \begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix}
= I
+ \begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix}
+ \frac{1}{2}\underbrace{\begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix}^2}_{=0} + \cdots
= \begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}.
</math>
== A functional analysis perspective ==
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