Reaching all of R^N by rotations from a linear subspace

In summary, the speaker is seeking to prove or disprove the possibility of reaching any point in ##\mathbb{R}^N## through rotations from a linear subspace ##L## of dimension ##N-3##. They propose using three matrices ##T_i## to generate these rotations and plan to add more details if needed. The expert summarizer then explains that any given vector ##v## can be transformed into a vector in ##L## through a rotation, and this can be extended to transform ##v## into any point in ##\mathbb{R}^N##.
  • #1
uekstrom
8
0
Hi all,

I have a problem related to quantum mechanical description of vibrational motion in molecules. I would like, for efficiency, to integrate over symmetries (global rotations) of the molecule.

I would like to prove (or disprove) that all points in [tex]R^N[/tex] can be reached by rotations from a linear subspace (let's call it L). In my particular case L has dimension N-3, and my rotations are generated by three matrices [tex]T_i[/tex], of the form

[tex] T_i = \mathrm{diag}(L_i,L_i,..) [/tex],

where [tex]L_x,L_y,L_z[/tex] are the 3x3 generators of SO(3). [tex]T_x[/tex] is then a block-diagonal matrix with copies of [tex]L_x[/tex] on the diagonal.

So what I want to prove (or disprove) is if, for a given x,

[tex] x = exp\left( c_x T_x + c_y T_y + c_z T_z \right) y [/tex]

has a solution for x from [tex]R^N[/tex] and y from the linear subspace L. L can be chosen arbitrarily, but should not depend on x. From numerical experiments is seems like this rotation is always possible, but I would like to have a formal proof. I will add more details if necessary.
 
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  • #2
Let ##v\in \mathbb{R}^N## be any given vector. Since ##L## is a linear subspace, there is an element in ##w\in L## with ##|v|=|w|##. Then there is always a rotation which transforms ##w## into ##v##. It can be solved in the two dimensional subspace spanned by ##v,w## and then extended in any way to ##\mathbb{R}^N##.
 

Related to Reaching all of R^N by rotations from a linear subspace

What is the concept of "Reaching all of R^N by rotations from a linear subspace"?

This concept refers to the idea that any point in a higher-dimensional space, denoted as R^N, can be reached by performing rotations from a lower-dimensional linear subspace. In other words, through a combination of rotations around specific axes, any point in R^N can be obtained from a starting point in a linear subspace.

What is the significance of this concept in mathematics?

This concept is significant in mathematics as it demonstrates the power and flexibility of rotations in higher-dimensional spaces. It also highlights the idea that a lower-dimensional subspace can serve as a basis for exploring and understanding higher-dimensional spaces.

How does this concept relate to other mathematical concepts?

This concept is closely related to the concept of vector spaces and linear transformations. It also has connections to group theory and symmetry, as rotations can be seen as a type of transformation that preserves certain properties of an object.

What are some practical applications of this concept?

This concept has practical applications in fields such as computer graphics, robotics, and physics. In computer graphics, it is used to represent and manipulate objects in three-dimensional space. In robotics, it is used to model and control the movements of robotic arms. In physics, it is used to describe the rotations of objects in space.

Are there any limitations to this concept?

While this concept is powerful and useful, it does have some limitations. It only applies to rotations and does not take into account other types of transformations. Additionally, it may not be applicable in situations where there are constraints or restrictions on the rotations that can be performed.

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