Editing Rotation matrix (section)

=== Independent planes ===
Consider the {{nowrap|3 × 3}} rotation matrix
:<math> Q = \begin{bmatrix} 0.36 & 0.48 & -0.80 \\ -0.80 & 0.60 & 0.00 \\ 0.48 & 0.64 & 0.60 \end{bmatrix} . </math>
If {{math|''Q''}} acts in a certain direction, {{math|'''v'''}}, purely as a scaling by a factor {{mvar|λ}}, then we have
:<math> Q \mathbf{v} = \lambda \mathbf{v}, </math>
so that
:<math> \mathbf{0} = (\lambda I - Q) \mathbf{v} . </math>
Thus {{mvar|λ}} is a root of the [[characteristic polynomial]] for {{mvar|Q}},
:<math>\begin{align}
 0 &{}= \det (\lambda I - Q) \\
   &{}= \lambda^3 - \tfrac{39}{25} \lambda^2  + \tfrac{39}{25} \lambda - 1 \\
   &{}= (\lambda-1) \left(\lambda^2 - \tfrac{14}{25} \lambda + 1\right).
\end{align}</math>
Two features are noteworthy. First, one of the roots (or [[eigenvalue]]s) is 1, which tells us that some direction is unaffected by the matrix. For rotations in three dimensions, this is the ''axis'' of the rotation (a concept that has no meaning in any other dimension). Second, the other two roots are a pair of complex conjugates, whose product is 1 (the constant term of the quadratic), and whose sum is {{math|2 cos ''θ''}} (the negated linear term). This factorization is of interest for {{nowrap|3 × 3}} rotation matrices because the same thing occurs for all of them. (As special cases, for a null rotation the "complex conjugates" are both 1, and for a 180° rotation they are both −1.) Furthermore, a similar factorization holds for any {{math|''n'' × ''n''}} rotation matrix. If the dimension, {{mvar|n}}, is odd, there will be a "dangling" eigenvalue of 1; and for any dimension the rest of the polynomial factors into quadratic terms like the one here (with the two special cases noted). We are guaranteed that the characteristic polynomial will have degree {{mvar|n}} and thus {{mvar|n}} eigenvalues. And since a rotation matrix commutes with its transpose, it is a [[normal matrix]], so can be diagonalized. We conclude that every rotation matrix, when expressed in a suitable coordinate system, partitions into independent rotations of two-dimensional subspaces, at most {{math|{{sfrac|''n''|2}}}} of them.

The sum of the entries on the main diagonal of a matrix is called the [[trace (linear algebra)|trace]]; it does not change if we reorient the coordinate system, and always equals the sum of the eigenvalues. This has the convenient implication for {{nowrap|2 × 2}} and {{nowrap|3 × 3}} rotation matrices that the trace reveals the [[angle of rotation]], {{mvar|θ}}, in the two-dimensional space (or subspace). For a {{nowrap|2 × 2}} matrix the trace is {{math|2 cos ''θ''}}, and for a {{nowrap|3 × 3}} matrix it is {{math|1 + 2 cos ''θ''}}. In the three-dimensional case, the subspace consists of all vectors perpendicular to the rotation axis (the invariant direction, with eigenvalue 1). Thus we can extract from any {{nowrap|3 × 3}} rotation matrix a rotation axis and an angle, and these completely determine the rotation.