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the series for log(1 + x) converges if |x| < 1 |
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:<math>\ln B=\ln \big(\lambda(I+K)\big)=\ln (\lambda I) +\ln (I+K)= (\ln \lambda) I + K-\frac{K^2}{2}+\frac{K^3}{3}-\frac{K^4}{4}+\cdots</math>
This [[series (mathematics)|series]] in general does not converge for any matrix ''K'', as it would not for any real number with absolute value greater than unity, however, this particular ''K'' is a [[nilpotent]] matrix, so the series actually has a finite number of terms (''K''<sup>''m''</sup> is zero if ''m'' is the dimension of ''K'').
Using this approach one finds
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