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→The logarithm of a non-diagonalizable matrix: Expand example to show that the given matrix is actually the logarithm. |
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==Power series expression==
If ''B'' is sufficiently close to the identity matrix, then a logarithm of ''B'' may be computed by means of the following [[power series]]:
:<math>\log(B)= \sum_{k=1}^\infty{(-1)^{k+1}\frac{(B-I)^k}{k}} =(B-I)-\frac{(B-I)^2}{2}+\frac{(B-I)^3}{3}-\frac{(B-I)^4}{4}+\cdots</math>.
Specifically, if <math>\left\|B-I\right\|<1</math>, then the preceding series converges and <math>e^{\log(B)}=B</math>.<ref>{{harvnb|Hall|2015}} Theorem 2.8</ref>
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