Is anyone's symbolic math package able to take the matrix exponential of [ 0 a 0 -d ] [ -a 0 b 0 ] [ 0 -b 0 c ] [ d 0 -c 0 ] ? I set Macsyma to work on it, and it seems only to be trying to melt my laptop and burn the house down after long enough to suspect it isn't going to finish. If a result of that is available, I'd also like to know the resulting eigenvalues and eigenvectors. Thanks in advance, Shel
Consider, more generally, [ 0 a -b c] [-a 0 d e] = M. [ b -d 0 f] [-c -e -f 0] Maple calculates the characteristic polynomial to be x^4 + (aa+bb+cc+dd+ee+ff) x^2 + (af+cd+be)^2 = 0. The eigenvalues are +iA, -iA, +iB, -iB, with A and B messy real-valued expressions (assuming a, ..., f are real). R = exp(M) is a 4-dimensional rotation. There is a pair of orthogonal planes P1 and P2 (determined by the eigenvectors of M) such that R is the commutative product of a rotation by angle A in P1 with P2 fixed, and a rotation by angle B in P2 with P1 fixed. --- Shel Kaphan <sjk@kaphan.org> wrote:
Is anyone's symbolic math package able to take the matrix exponential of
[ 0 a 0 -d ] [ -a 0 b 0 ] [ 0 -b 0 c ] [ d 0 -c 0 ]
?
I set Macsyma to work on it, and it seems only to be trying to melt my laptop and burn the house down after long enough to suspect it isn't going to finish.
If a result of that is available, I'd also like to know the resulting eigenvalues and eigenvectors.
Thanks in advance,
Shel
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Thanks. (and also to Dan A.) I guess I owe a little bit of explanation as to why I even cared. I was looking for the simplest case of a 4-d rotation that would not "trivially" be reducible to a lower dimensional case. (Interestingly, it is exactly this difference that seems to throw Macsyma for a loop on this). Now I see (thanks to some offline comments from Dan) that any rotation in R^n is representable as some number of orthogonal 2-d rotations (leaving out many details). One thing I found interesting to try to understand is the meaning of the elements of a skew-symmetric matrix in relation to the rotation matrix obtained by exponentiating it. In 3-d, they represent a vector giving the axis of rotation, and the amount of rotation (the length of that vector). But in higher dimensions, there is no axis. It appears the (i,j) element represents the amount of rotation in the (i,j) coordinate plane. As there are (n choose 2) (i,j) planes, this is an easy way to see that the dimension of SO(n) is (n choose 2). The total rotation doesn't necessarily take the coordinate planes into themselves, due to interaction between rotations in coordinate planes with a common coordinate. You could consider a skew-symmetric matrix as the sum of a number of skew-symmetric matrices with only one pair of non-zero elements each. These each represent a simple rotation in one coordinate plane. These component rotations are combined when exponentiating to get the corresponding rotation matrix, using (also from Dan): exp(a+b+...) = lim[n->oo] (exp(a)^(1/n) exp(b)^(1/n) ...) ^n, or (I suppose) using the messy BCH stuff. Either way it is a commutative combination of the component rotations that, e.g, in 3-d is equivalent to vector addition of axial vectors of the component rotations. So, with some change of basis, an arbitrary rotation in R^n should be representable by a skew-symmetric matrix with only max(floor(n/2)) non-zero components, arranged so that the rotations are in independent coordinate planes, i.e. they have no shared coordinate. Shel At 01:05 PM 7/14/2003 -0700, Eugene Salamin wrote:
Consider, more generally,
[ 0 a -b c] [-a 0 d e] = M. [ b -d 0 f] [-c -e -f 0]
Maple calculates the characteristic polynomial to be
x^4 + (aa+bb+cc+dd+ee+ff) x^2 + (af+cd+be)^2 = 0.
The eigenvalues are +iA, -iA, +iB, -iB, with A and B messy real-valued expressions (assuming a, ..., f are real). R = exp(M) is a 4-dimensional rotation. There is a pair of orthogonal planes P1 and P2 (determined by the eigenvectors of M) such that R is the commutative product of a rotation by angle A in P1 with P2 fixed, and a rotation by angle B in P2 with P1 fixed.
--- Shel Kaphan <sjk@kaphan.org> wrote:
Is anyone's symbolic math package able to take the matrix exponential of
[ 0 a 0 -d ] [ -a 0 b 0 ] [ 0 -b 0 c ] [ d 0 -c 0 ]
?
I set Macsyma to work on it, and it seems only to be trying to melt my laptop and burn the house down after long enough to suspect it isn't going to finish.
If a result of that is available, I'd also like to know the resulting eigenvalues and eigenvectors.
Thanks in advance,
Shel
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Quoting Shel Kaphan <sjk@kaphan.org>:
Thanks. (and also to Dan A.)
I guess I owe a little bit of explanation as to why I even cared. I was looking for the simplest case of a 4-d rotation that would not "trivially" be reducible to a lower dimensional case. (Interestingly, it is exactly this difference that seems to throw Macsyma for a loop on this). Now I see (thanks to some offline comments from Dan) that any rotation in R^n is representable as some number of orthogonal 2-d rotations (leaving out many details).
One thing I found interesting to try to understand is the meaning of the elements of a skew-symmetric matrix in relation to the rotation matrix obtained by exponentiating it. In 3-d, they represent a vector giving the axis of rotation, and the amount of rotation (the length of that vector). But in higher dimensions, there is no axis. It appears the (i,j) element represents the amount of rotation in the (i,j) coordinate plane. As there are (n choose 2) (i,j) planes, this is an easy way to see that the dimension of SO(n) is (n choose 2). The total rotation doesn't necessarily take the coordinate planes into themselves, due to interaction between rotations in coordinate planes with a common coordinate. You could consider a skew-symmetric matrix as the sum of a number of skew-symmetric matrices with only one pair of non-zero elements each. These each represent a simple rotation in one coordinate plane. These component rotations are combined when exponentiating to get the corresponding rotation matrix, using (also from Dan): exp(a+b+...) = lim[n->oo] (exp(a)^(1/n) exp(b)^(1/n) ...) ^n, or (I suppose) using the messy BCH stuff. Either way it is a commutative combination of the component rotations that, e.g, in 3-d is equivalent to vector addition of axial vectors of the component rotations.
So, with some change of basis, an arbitrary rotation in R^n should be representable by a skew-symmetric matrix with only max(floor(n/2)) non-zero components, arranged so that the rotations are in independent coordinate planes, i.e. they have no shared coordinate.
Shel
I shouldn't have copied all this, but it is useful to have it on hand to be able to comment on it. In formulating a response to a question, it helps to know the context and motivation of the question. Giving a symbol manipulation program a hard time invites one response; to know about the structure of rotation groups invites another. We have had previous discussions of such, although I cannot give chapter and verse right off. An interesting intersection of these two trains of thought is the fact that the Lie Algebra of O4 is the direct product of the Lie Algebras of two O3 rotation groups, something which generalizes to the Lorentz group but not elsewhere. With this in mind, write the 4x4 antisymmetric matrix as | 0 Ez -Ey Bx | | -Ez 0 Ex By | | Ey -Ex 0 Bz | | -Bx -By -Bz 0 | where the similarity of E and B to the symbols for electric and magnetic field strength is no accident. Then (Ex,Ey, Ez) defines the axis of one rotation about an angle which is the length of the vector. Two of these rotations do not commute, but the Campbell-Hausdorff formula has a nice interpretation in this special case. Represent the rotations os arcs of great circles; their poles are the axes of rotation, and the arcs are the angles of rotation (on the unit sphere, of course). These arcs add vectorially, tip to tail, and the non- commutativity is due to the curvature of the spherical surface (hyperbolic if need be for the Lorentz group). The B vector (Bx, By, Bz) describes the other rotation. Different coordinate choices for the representative sphere can give angle-axis (described above), Euler angles, or quaternions. Looking around the literature there are lots of places this is neatly described, including how to factor a rotation into two refelections. So if your prime interest is in 3 and 4 dimensional rotations, you can leave your poor little laptop computer in peace; or better yet, connect it to the Internet and do a Google search. Good luck! - hvm ------------------------------------------------- Obtén tu correo en www.correo.unam.mx UNAMonos Comunicándonos
participants (3)
-
Eugene Salamin -
mcintosh＠servidor.unam.mx -
Shel Kaphan