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In each coordinate space $V$ with dimension $\dim(V)$, we can describe any rotation operator $R : V \to V$ as a product of rotations in as few as $\dim(V) - 1$ orthogonal planes in the space.

Let's say that given the canonical basis $\mathcal B$ of $V$, we take the canonical ordering of the basis vectors $\vec b_i$ and consider only rotations in the planes $\mathcal P = \{\ \mathcal P_{i, i+1} = \text{span} (b_i,b_{i+1})\ |\ b_i, b_{i+1} \in \mathcal B\ \}$. Is there a closed form solution for the matrix of the final rotation operator?

Consider an example in three dimensions. Let $R_{i, j}(\theta_k)$ be the rotation by $\theta_k$ in the plane $\mathcal P_{i, j}$.

\begin{align} R(\theta_1, \theta_2) = R_{1,2}(\theta_1)R_{2,3}(\theta_2) =& \left[\begin{array}{ccc} \cos(\theta_1) & -\sin(\theta_1) & 0 \\ \sin(\theta_1) & \cos(\theta_1) & 0 \\ 0 & 0 & 1 \end{array}\right] \left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & \cos(\theta_2) & -\sin(\theta_2) \\ 0 & \sin(\theta_2) & \cos(\theta_2) \end{array}\right] \\ =& \left[\begin{array}{ccc} \cos(\theta_1) & -\sin(\theta_1)\cos(\theta_2) & \sin(\theta_1)\sin(\theta_2) \\ \sin(\theta_1) & \cos(\theta_1)\cos(\theta_2) & -\cos(\theta_1)\sin(\theta_2) \\ 0 & \sin(\theta_2) & \cos(\theta_2) \end{array}\right] \end{align}

I can see the beginnings of a pattern, but I feel as though the pattern will break shortly after increasing the number of dimensions.

Is there already a well-studied closed form for this? Is it a fool's errand?

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2 Answers 2

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Maybe this ?

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It seems that you get a Hessenberg matrix with all sines on the subdiagonal, cosine pairs on the main diagonal, superdiagonals with alternating signs and products on increasing lengths, not so easy to describe.


Update:

With better alignment and pseudo-factors $c_{\bar1},c_4$, the pattern becomes much clearer.

The subdiagonal has all sines. The upper triangle has pairs of cosines (extended row and column indexes), and suffix products of the sines.

$$\left(\begin{matrix} &\color{green}{c_{\bar1}}\color{blue}{c_0}&-\color{green}{c_{\bar1}}\color{blue}{c_1}s_0&\ \ \ \color{green}{c_{\bar1}}\color{blue}{c_2}s_0s_1&-\color{green}{c_{\bar1}}\color{blue}{c_3}s_0s_1s_2&\ \ \ \color{green}{c_{\bar1}}\color{blue}{c_4}s_0s_1s_2s_3\\ &\color{magenta}{s_0}&\ \ \ \color{green}{c_0}\color{blue}{c_1}\ \ \ &-\color{green}{c_0}\color{blue}{c_2}\ \ \ \ s_1&\ \ \ \color{green}{c_0}\color{blue}{c_3}\ \ \ \ s_1s_2&-\color{green}{c_0}\color{blue}{c_4}\ \ \ \ s_1s_2s_3\\ &0&\color{magenta}{s_1}&\ \ \ \color{green}{c_1}\color{blue}{c_2}\ \ \ \ \ \ \ \ &-\color{green}{c_1}\color{blue}{c_3}\ \ \ \ \ \ \ \ s_2&\ \ \ \color{green}{c_1}\color{blue}{c_4}\ \ \ \ \ \ \ s_2s_3\\ &0&0&\color{magenta}{s_2}&\ \ \ \color{green}{c_2}\color{blue}{c_3}\ \ \ \ \ \ \ \ \ \ \ &-\color{green}{c_2}\color{blue}{c_4}\ \ \ \ \ \ \ \ \ \ \ s_3\\ &0&0&0&\color{magenta}{s_3}&\color{green}{c_3}\color{blue}{c_4}\ \ \ \ \ \ \ \ \ \ \ \\ \end{matrix}\right)$$

For the upper triangular elements,

$$R_{rc}=(-1)^{r+c}\color{green}{\cos(\theta_{r-1})}\color{blue}{\cos(\theta_{c})}\prod_{k=r}^{c-1}\sin(\theta_k).$$

By computing the products incrementally, the whole matrix can be obtained in $\approx\dfrac32d^2$ multiplies.

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  • $\begingroup$ I think this answer is more of a push in the right direction. If this could be generalized, it's exactly what I'm looking for. $\endgroup$
    – Axoren
    Dec 12, 2015 at 1:00
  • $\begingroup$ When looking from top-right, you see triangles of sines decreasing in size. And a little more regularity by introducing $c_{-1}=c_{4}=1$. $\endgroup$
    – user65203
    Dec 12, 2015 at 10:58
  • $\begingroup$ I've been mulling it over for the past couple of days. By your selection of planes in which the rotation is done, the pattern becomes nearly predictable. It looks like it becomes an exercise in permutations which could be more efficient to compute than the product of the matrices independently. The fact that it produces a Hessenberg Matrix is a major boon. I can already foresee a dynamic programming approach to solving this if this pattern persists past $\mathbb R ^5$ $\endgroup$
    – Axoren
    Dec 13, 2015 at 5:40
  • $\begingroup$ @Axoren: the pattern must be incremental, from dimension $d$ to $d+1$. $\endgroup$
    – user65203
    Dec 13, 2015 at 13:46
  • $\begingroup$ I'm inclined to agree now. I checked it earlier this morning. The pattern does persist. It continues to be Hessenburg for all $d$ with a predictable pattern. What you mentioned before about considering the $c_4 = 1$ is actually the basis for inducting it's completeness to higher dimensions. Were $R$ a transformation in a higher dimensional space that only rotated a lower dimensional subspace. The planes in dimensions it doesn't touch would have $\text{cosines}$ of $1$ and we can say this for all higher dimensional spaces that could contain this rotation in a subspace. $\endgroup$
    – Axoren
    Dec 13, 2015 at 20:14
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To represent rotation matrices nicely, we can use the Jordan normal form.

A rotation operator can be represented by an orthogonal matrix (a matrix $R$ satisfying $R^T R = I = R R^T$). For these matrices, we have the result that their (complex) eigenvalues satisfy $|\lambda|^2 = 1$ (you can prove this by calculating $||\lambda v||^2$ for some eigenvector $v$). This means that either the eigenvalues are $1$ or $-1$, or they are complex. For complex eigenvalues we can write $\lambda_i = e^{i\theta_i} = \cos\theta_i + i \sin\theta_i$.

We can then use the real Jordan form to see that $R$ is similar to: $$ RS = SJ\\[1cm] J = \left[ \begin{array}{ccccc|ccc|ccc} \cos\theta_1 & \sin\theta_1 \\ -\sin\theta_1 & \cos\theta_1 \\ & & \ddots \\ & & & \cos \theta_m & \sin\theta_m\\ & & & -\sin\theta_m & \cos\theta_m\\\hline &&&&& -1\\ &&&&&& \ddots\\ &&&&&&& -1\\\hline &&&&&&&& 1\\ &&&&&&&&& \ddots\\ &&&&&&&&&& 1 \end{array} \right] $$ (The fact that no identity matrices occur on the upper diagonal is due to the fact that an orthogonal matrix can be diagonalized.) Here $2m$ is the number of non-real eigenvalues (note that these always occur in complex conjugate pairs, which allows this representation). The matrix that transforms $R$ into this form has the real and imaginary parts of the eigenvectors of $R$ as its columns (in the same order as the eigenvalues in $J$). For a proper rotation, the number of $-1$ eigenvalues will be even.

Hence, we can see that a general rotation operator can be represented as $m$ rotations that influence only a 2-dimensional space (spanned by the real and complex part of an eigenvector corresponding to the complex eigenvalue), and an even number of reflections in some hyperplanes (orthogonal to the eigenvector corresponding to the $-1$ eigenvalues).

Clarification

In response to the comments I would like give some additional clarifications to my answer, hoping to shed light on the difference between the two Jordan forms and the interpretation of the result.

The Jordan Normal Form

Let $M \in \mathbb{R}^{n \times n}$ be a matrix with real entries. In general it will then have $m$ distinct eigenvalues $\lambda_i \in \mathbb{C}$. The matrix might have enough eigenvectors to form a basis $S = [\vec{v_1^{(1)}},\dots,\vec{v_{k_1}^{(1)}},\dots,\vec{v_1^{(m)}},\dots,\vec{v_{k_m}^{(m)}}]$. In this case you can diagonalize the matrix by $D = S^{-1} M S$. However, in general this is not possible. In this case you have to build a number of Jordan chains for each eigenvalue.

We let $k_i$ denote the number of Jordan chains corresponding to $\lambda_i$ and $l_{ij}$ the length of the $j$th Jordan chain of $\lambda_i$. We then have a basis of generalized eigenvectors (for $j = 1,\dots,k_i$): $$ (A - \lambda_i I) \vec{v_{j, p}^{(i)}} = \vec{v_{j,p-1}^{(i)}}, \qquad p = l_{ij},\dots,2\\ (A - \lambda_i I) \vec{v_{j, 1}^{(i)}} = 0 $$ We can now arrange these in a matrix: $$ S = [ \vec{v_{1,1}^{(1)}},\dots,\vec{v_{1,l_{11}}^{(1)}},\dots, \vec{v_{k_1,1}^{(1)}},\dots,\vec{v_{k_1,l_{k_1 1}}^{(1)}},\dots, \vec{v_{1,1}^{(m)}},\dots,\vec{v_{1,l_{1 m}}^{(m)}},\dots, \vec{v_{k_m,1}^{(m)}},\dots,\vec{v_{k_m,l_{m k_m}}^{(m)}} ] $$ We then see that $M$ takes on the following form: $$ J = S^{-1} M S = \left[\begin{array}{} J_1 \\ & J_2\\ && \ddots\\ &&& J_m \end{array}\right], \quad\text{where:} \\ J_i = \left[\begin{array}{} J_{i1}\\ & \ddots\\ && J_{ik_i} \end{array}\right] \quad\text{and}\quad J_{ij} = \left[\begin{array}{} \lambda_i & 1\\ & \lambda_i & \ddots \\ & & \ddots & 1 \\ & & & \lambda_i \end{array}\right] \in \mathbb{C}^{l_{ij} \times l_{ij}}. $$ This is then finally called the Jordan normal form.

Real Jordan form

We now make the powerful observation that if $\lambda$ is an eigenvalue of $M$ with a Jordan chain $\vec{v_1},\dots,\vec{v_l}$, then the complex conjugate $\lambda^*$ is also an eigenvalue with Jordan chain $\vec{v_1}^*,\dots,\vec{v_l}^*$. Hence, for non-real eigenvalues, we can take a linear combination of these chains to form the real and imaginary parts.

To this end, let us denote the number of non-real eigenvalues bij $2\tilde{m}$ (this is always even since they occur in pairs) and the number of real eigenvalues by $m$. We then reorder the eigenvalues and Jordan chains above so that no pairs of complex conjugate eigenvectors occur as the first $\tilde{m}$, and define: $$ a_i = \Re(\lambda_i) \quad b_i = \Im(\lambda_i) \\ \vec{v^{(i)R}_{jp}} = \Re(\vec{v^{(i)}_{jp}}) \quad \vec{v^{(i)I}_{jp}} = \Im(\vec{v^{(i)}_{jp}}) $$ If you then work out the formulas, you get: $$ M\vec{v^{(i)R}_{jp}} = a_i \vec{v^{(i)R}_{jp}} - b_i \vec{v^{(i)I}_{jp}} + \vec{v^{(i)R}_{j (p-1)}}\\ M\vec{v^{(i)I}_{jp}} = b_i \vec{v^{(i)R}_{jp}} + a_i \vec{v^{(i)I}_{jp}} + \vec{v^{(i)I}_{j (p-1)}} $$ Hence, setting the real eigenvalues to $c_i$ with Jordan chains $w^{(i)}_{jp}$ and redefining the matrix S to: \begin{multline} S = [ \vec{v^{(1)R}_{11}},\vec{v^{(1)I}_{11}},\dots, \vec{v^{(1)R}_{1 l_{11}}},\vec{v^{(1)I}_{1 l_{11}}},\dots, \vec{v^{(\tilde{m})R}_{k_\tilde{m} 1}},\vec{v^{(\tilde{m})I}_{k_\tilde{m} 1}},\dots, \vec{v^{(\tilde{m})R}_{k_\tilde{m} l_{\tilde{m} k_\tilde{m}}}},\vec{v^{(\tilde{m})I}_{k_\tilde{m} l_{\tilde{m} k_\tilde{m}}}}, \\\dots, \vec{w^{(1)}_{11}},\dots,\vec{w^{(m)}_{k_m l_{m k_m}}} ] \end{multline} we now see that we get the following matrix: $$ J = S^{-1} M S = \left[\begin{array}{ccc|ccc} K_1\\ & \ddots\\ && K_\tilde{m}\\ \hline &&& J_1\\ &&&& \ddots\\ &&&&& J_m \end{array}\right] \text{, where:}\quad K_i = \left[\begin{array}{} K_{i1}\\ & \ddots\\ && K_{i k_i} \end{array}\right] \text{ and}\\ K_{ij} = \left[\begin{array}{cc|cc|cc|cc} a_i & b_i & 1 & 0\\ -b_i & a_i & 0 & 1\\ \hline && a_i & b_i & \ddots\\ && -b_i & a_i && \ddots\\ \hline &&&& \ddots && 1 & 0\\ &&&&& \ddots & 0 & 1\\ \hline &&&&&& a_i & b_i\\ &&&&&& -b_i & a_i \end{array}\right] \in \mathbb{R}^{2l_{ij} \times 2l_{ij}} $$ and the $J_i$ are as above.

Orthogonal matrices

As stated above, the nice thing about orthogonal matrices is that their eigenvalues satisfy $|\lambda|^2 = 1$ and that all Jordan chains are of length 1. Hence you can always write $\lambda = e^{i\theta} = \cos\theta + i \sin\theta$ for some $\theta \in \mathbb{R}$ and find a basis of eigenvectors. This allows you to interpret an orthogonal matrix in a very straightforward way, namely as a series of operations on independent 1- or 2-dimensional subspaces:

  • For a non-real eigenvalue the real and imaginary part of an eigenvector span a 2-dimensional space on which the matrix operates as a rotation of an angle equal to the argument of the eigenvalue.
  • For a real eigenvalue the matrix acts either as the identity (1) or a reflection (-1) in the hyperplane normal to the corresponding eigenvector.
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  • $\begingroup$ Forgive me, I'm not as familiar with Block Matrices and Jordan normal form. Given a vector $\vec v \in V$, how would I apply $J$ to it? Are the 1s and -1s blocks or single elements? $\endgroup$
    – Axoren
    Dec 7, 2015 at 2:03
  • $\begingroup$ The Jordan normal form is simply a nice form a matrix takes on in a particular basis (consisting of eigenvectors and generalized eigenvectors). By the relation $R = SJS^{-1}$, applying $R$ to a vector $\vec{v}$ would be the same as first performing a change of basis with $S^{-1}$, then applying $J$ in that new, "nice", basis, and then changing back with $S$. The 1's and -1's are just single elements. I really advice you to read up on Jordan normal form. It is in my opinion one of the most powerful tools in real and complex linear algebra. $\endgroup$ Dec 10, 2015 at 2:52
  • $\begingroup$ As I've read up on Jordan normal form in the past couple of days, I've yet to see the contribution your answer has made. For that, I apologize. However, I do not see how this matrix $J$ is a Jordan matrix. It does not contain Jordan blocks along the diagonal. It instead contains rotation matrices along the diagonal (plus more reflections). $\endgroup$
    – Axoren
    Dec 10, 2015 at 3:03
  • $\begingroup$ One issue I have with this answer as it stands is that now I've got a larger problem: Assume that your $J$ matrix is correct and that it represents the rotation in another basis defined by $S$. I'm now lacking a closed form for $S$ which is in fact another matrix which may or may not be a rotation matrix. $\endgroup$
    – Axoren
    Dec 10, 2015 at 3:06
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    $\begingroup$ After playing around with this some more, I was also able to show that in fact all eigenvectors and the real linear combinations of an orthogonal matrix are orthogonal and can be chosen to be orthonormal. Hence, indeed $S$ will be another rotation matrix (possibly with a scaling). I now see what you are getting at. What will happen if you decompose $S$ in the same way?... $\endgroup$ Dec 10, 2015 at 12:09

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