Logarithm of a matrix: Difference between revisions

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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 [[power series]]

: <math>\log(B)=\log(I-(I-B))= -\sum_{k=1}^\infty{(-1)^{k+1} \frac{(I-B)^k}{k}}

</math>.

Specifically, if <math>\left\|I-B\right\|<1</math>, then the preceding series converges and <math>e^{\log(B)}=B</math>.<ref>{{harvnb|Hall|2015}} Theorem 2.8</ref>