Euler's Rotation Theorem - Equivalence of An Orthogonal Matrix To A Rotation Matrix

Equivalence of An Orthogonal Matrix To A Rotation Matrix

Two matrices (representing linear maps) are said to be equivalent if there is change of basis that makes one equal to the other. A proper orthogonal matrix is always equivalent (in this sense) to either the following matrix or to its vertical reflection:

\mathbf{R} \sim \begin{pmatrix} \cos\phi & -\sin\phi & 0 \\ \sin\phi & \cos\phi & 0 \\ 0 & 0 & 1\\ \end{pmatrix}, \qquad 0\le \phi \le 2\pi.

Then, any orthogonal matrix is either a rotation or an improper rotation. A general orthogonal matrix has only one real eigenvalue, either +1 or −1. When it is +1 the matrix is a rotation. When −1, the matrix is an improper rotation.

If R has more than one invariant vector then φ = 0 and R = E. Any vector is an invariant vector of E.

Read more about this topic:  Euler's Rotation Theorem

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