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XOR'easter (talk | contribs) →Constraints in the 2 × 2 case: remove unsourced section that relies on terminology which is, at best, niche Tag: Reverted |
→Example: Logarithm of rotations in the plane: Collapse Proof |
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</math>
is a logarithm of ''A''. <br>
is a logarithm of ''A''. Thus, the matrix ''A'' has infinitely many logarithms. This corresponds to the fact that the rotation angle is only determined up to multiples of 2''π''.▼
{{Collapse top|title=Proof}}
<math>
log(A) =B_n~</math>⇔<math>~~e^{B_n} =A
</math><br><br>
<math>
e^{B_n} = \sum_{k=0}^\infty{1 \over k!}B_n^k
~</math> where <br>
<math>
(B_n)^0=
1~I_2,
</math><br>
<math>
(B_n)^1=
(\alpha+2\pi n)\begin{pmatrix}
0 & -1 \\
+1 & 0\\
\end{pmatrix},
</math><br>
<math>
(B_n)^2=
(\alpha+2\pi n)^2\begin{pmatrix}
-1 & 0 \\
0 & -1 \\
\end{pmatrix},
</math><br>
<math>
(B_n)^3=
(\alpha+2\pi n)^3\begin{pmatrix}
0 & 1 \\
-1 & 0\\
\end{pmatrix},
</math><br>
<math>
(B_n)^4=
(\alpha+2\pi n)^4~I_2
</math><br>
… <br>
<math>
\sum_{k=0}^\infty{1 \over k!}B_n^k =\begin{pmatrix}
\cos(\alpha) & -\sin(\alpha) \\
\sin(\alpha) & \cos(\alpha) \\
\end{pmatrix} =A~.
</math>
<br>
qed.
{{Collapse bottom}}<br>
▲
In the language of Lie theory, the rotation matrices ''A'' are elements of the Lie group [[circle group|SO(2)]]. The corresponding logarithms ''B'' are elements of the Lie algebra so(2), which consists of all [[skew-symmetric matrix|skew-symmetric matrices]]. The matrix
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