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 - I)) = -\sum_{k=1}^{\infty} \frac{(I-B1)^{k + 1}}{k} (B - I)^k = (B - I) - \frac{(B - I)^2}{2} + \frac{(B - I)^3}{3} - \cdots</math>,

which can be rewritten as

</math>.

:<math>\log(B) = -\sum_{k=1}^{\infty} \frac{(I - B)^k}{k} = -(I - B) - \frac{(I - B)^2}{2} - \frac{(I - B)^3}{3} - \cdots</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>