Suppose [itex]X\in\mathbb{R}^{n\times n}[/itex] is orthogonal. How do you perform the computation of series(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\log(X) = (X-1) - \frac{1}{2}(X-1)^2 + \frac{1}{3}(X-1)^3 - \cdots

[/tex]

Elements of individual terms are

[tex]

((X-1)^n)_{ij} = (-1)^n\delta_{ij} \;+\; n(-1)^{n-1}X_{ij} \;+\; \sum_{k=2}^{n} (-1)^{n-k} \frac{n!}{k!(n-k)!} X_{il_1} X_{l_1 l_2} \cdots X_{l_{k-1}j}

[/tex]

but these do not seem very helpful. I don't see how orthogonality could be used here. By orthogonality we have

[tex]

X_{ik}X_{jk} = \delta_{ij},\quad\quad X_{ki}X_{kj} = \delta_{ij}

[/tex]

but

[tex]

X_{ik}X_{kj}

[/tex]

is nothing special, right?

The special case [itex]n=2[/itex] would also be nice to start with. If

[tex]

X = \left(\begin{array}{cc}

\cos(\theta) & -\sin(\theta) \\

\sin(\theta) & \cos(\theta) \\

\end{array}\right)

[/tex]

then the result should be

[tex]

\log(X) = \left(\begin{array}{cc}

0 & -\theta \\

\theta & 0 \\

\end{array}\right)

[/tex]

but how does one arrive at this honestly from the series?

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# How to compute logarithm of orthogonal matrix?

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