## Show orthogonal matrices are manifolds (Munkres Analysis on Manifolds)

1. The problem statement, all variables and given/known data
Let ##O(3)## denote the set of all orthogonal 3 by 3 matrices, considered as a subspace of ##\mathbb{R}^9##.
(a) Define a ##C^\infty## ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6## such that ##O(3)## is the solution set of the equation ##f(x) = 0##.
(b) Show that ##O(3)## is a compact 3-manifold in ##\mathbb{R}^9## without boundary. [Hint: Show the rows of ##Df(x)## are independent if ##X \in O(3)##.

2. Relevant equations
I think the following theorem:

Let ##O## be open in ##\mathbb{R}^n##; let ##f:O \rightarrow \mathbb{R}## be of class ##C^r##. Let ##M## be the set of points ##x## for which ##f(x) = 0##; let ##N## be the set of points for which ##f(x) \geq 0##. Suppose ##M## is non-empty and ##Df(x)## has rank 1 at each point of ##M##. Then ##N## is an n-manifold in ##\mathbb{R}^n## and the boundary of ##N## is ##M##.

might be useful, but I don't see an opening for how I can apply it to the problem.

3. The attempt at a solution
I'm stuck on part (a) of the question, I think because the construction of the function seems kind of open-ended to me, as well as unmotivated in relation to part (b) (e.g. Why ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6##?). I feel like the function satisfying the required conditions may not be unique, like maybe something like
##f(\textbf{x},\textbf{y},\textbf{z}) = (\textbf{x} \cdot \textbf{y}, \textbf{x} \cdot \textbf{z}, \textbf{y} \cdot \textbf{z}, \textbf{x} \cdot \textbf{y}, \textbf{x} \cdot \textbf{z}, \textbf{y} \cdot \textbf{z} ) ##
or some permutation of it will satisfy the conditions for ##f## but is not unique (where ##\textbf{x}, \textbf{y} , \textbf{z}## are the orthogonal vectors making up the orthogonal matrix). Am I completely off the mark? I'm quite lost on this problem, any help would be greatly appreciated!

 PhysOrg.com science news on PhysOrg.com >> 'Whodunnit' of Irish potato famine solved>> The mammoth's lament: Study shows how cosmic impact sparked devastating climate change>> Curiosity Mars rover drills second rock target

 Quote by mathmonkey ...(a) Define a ##C^\infty## ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6## such that ##O(3)## is the solution set of the equation ##f(x) = 0##... ...3. The attempt at a solution I'm stuck on part (a) of the question, I think because the construction of the function seems kind of open-ended to me, as well as unmotivated in relation to part (b) (e.g. Why ##f:\mathbb{R}^9 \rightarrow \mathbb{R}^6##?). I feel like the function satisfying the required conditions may not be unique, like maybe something like ##f(\textbf{x},\textbf{y},\textbf{z}) = (\textbf{x} \cdot \textbf{y}, \textbf{x} \cdot \textbf{z}, \textbf{y} \cdot \textbf{z}, \textbf{x} \cdot \textbf{y}, \textbf{x} \cdot \textbf{z}, \textbf{y} \cdot \textbf{z} ) ## or some permutation of it will satisfy the conditions for ##f## but is not unique (where ##\textbf{x}, \textbf{y} , \textbf{z}## are the orthogonal vectors making up the orthogonal matrix). Am I completely off the mark? I'm quite lost on this problem, any help would be greatly appreciated!
I had to stare at it a while, because at first I was pulled towards the whole det(X)=1 idea. But I think your idea is good, the columns should be orthogonal. What else should they be?

 Hi algebrat, Thanks again for helping! I'm not sure what other properties our function should have. I suppose we can make each component function independent as well, so that our jacobian will have full rank? So maybe the function could be like: ##f(\textbf{x}, \textbf{y}, \textbf{z}) = (\textbf{x} \cdot \textbf{y}, \textbf{x} \cdot \textbf{z}, \textbf{y} \cdot \textbf{z}, \textbf{x} \cdot (\textbf{y} + \textbf{z}), \textbf{y} \cdot (\textbf{x} + \textbf{z}), \textbf{z} \cdot (\textbf{x} + \textbf{y})) ##? But my bigger problem is I still don't understand why we're trying to construct this function? How does it help with part (b) of the problem? Especially since all the dimensions are different. We're trying to show ##O(3)## is a 3-manifold but we're constructing a function whose range is ##\mathbb{R}^6##?

## Show orthogonal matrices are manifolds (Munkres Analysis on Manifolds)

 Quote by algebrat I had to stare at it a while, because at first I was pulled towards the whole det(X)=1 idea. But I think your idea is good, the columns should be orthogonal. What else should they be?
What additional condition must the columns satisfy? Find some examples of O(n) for various spaces.

There's one more conceptually/geometrically understandable fact about them you can know, besides just being orthogonal to each other. What is it?

 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus A tiny bit more help: what is $\mathbf{x}\cdot \mathbf{x}$ for columns $\mathbf{x}$??
 Hi, Hmm, I see! So since the vectors making up the matrix are orthonormal, we should also have ##||\textbf{x}||^2 = ||\textbf{y}||^2 = ||\textbf{z}||^2 = 1##. So, here's another stab at it: ##f(\textbf{x},\textbf{y},\textbf{z}) = (||\textbf{x}||^2 - 1 , ||\textbf{y}||^2 - 1, ||\textbf{z}||^2 - 1, \textbf{x} \cdot \textbf{ y}, \textbf{x} \cdot \textbf{ z}, \textbf{y} \cdot \textbf{ z}) ## So the advantage of this function now is ##f(A)=0## if and only if ##A \in O(3)##. Before in my previous example, the matrix ##\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}## would have satisfied the conditions required, but ##A## is clearly not an orthogonal matrix. Was this along the lines of what you two were hinting at? I guess I should have realized that sooner. Now, I believe the function satisfies the requirements for part (a) since ##f## is also ##C ^ \infty##. However, the question remains of how do I relate this result to part (b)? I'm thinking that I want to somehow transform this function so that I can apply the theorem in the original post, but the problem is the theorem requires a real-valued function, while I have one whose range is ##\mathbb{R}^6## (Maybe I can just define a new function ##g## as the sum of the component functions of ##f##?), along with other problems. Any more suggestions or hints for how I should proceed? Thanks again for all the help!
 Blog Entries: 8 Recognitions: Gold Member Science Advisor Staff Emeritus Use exercise 2 of the same chapter.

 Quote by micromass Use exercise 2 of the same chapter.
Which book are y'all using.

Blog Entries: 8
Recognitions:
Gold Member