Hello everyone,


I have a very interesting question on orthogonal projection matrices. Intuitively it is quite straightforward to understand. But for me it is not easy to prove.


In $R^2$ space, $a\_i$, $i=1,..,n>2$ are unit verctors, which are located on the unit circle uniformly. That means the angle subtended by any adjacent vectors is $2\pi/n$. It is known $A\_i=a\_ia\_i^T$ is an orthogonal projection matrix satisfying $A\_i^T=A\_i$ and $A\_i^2=A\_i$.


Prove: $A\_1+..+A\_n=a\_1a\_1^T+...+a\_na\_n^T=n/2I$ where $I$ is the identity matrix.


Remark:
1. Intuitively, consider three vectors. If their sum is zero, the vectors are distributed on the unit circle uniformly. 2. Guess: In $R^d$ space instead of $R^2$, the 1/2 in the equation should be 1/d. 3. My approach: prove $B=a\_1a\_1^T+...+a\_na\_n^T$ equals $n/2I$ if and only if $Bx=n/2x$ for arbitrary $x$ in $R^2$. Maybe a bad approach.


Thank you very much!


Shiyu


PS: consider the converse question:
if $a\_1a\_1^T+...+a\_na\_n^T=n/2I$, can we say $a\_1+...+a\_n=0$? 
Answer: using Nick's method, it can be shown this converse statement does not hold. Please refer to Nick's solution.


Now I'd like to pose a more general question. 


If $\sum k\_ia\_i=0$ where $k\_i>0$ and $\sum k\_i=1$, then $ \sum k\_ia\_ia\_i^T=1/dI$ where $d$ is the dimension of the space $R^d$.


The original problem in fact is a special case of the above statement where $k\_i=1/n$. Of course, this above statement is wrong becasue I must add some hypothesis like the unit vectors $a\_i$ should be located on the unit circle uniformly. But I am not sure how to state it rigorously at present. 



Prove: if a1,...,an are uniformly distributed unit vectors, then a1*a1'+...+an*an'=n/2*I
OK, here might be an answer to the question you are meaning to ask:


Let $a\_1$, ..., $a\_n$ be unit vectors in $\mathbb{R}^d$. Let $G$ be a group acting linearly on $\mathbb{R}^d$, which permutes the $a\_i$, and such that the representation $\mathbb{R}^d$ of $G$ is irreducible. For example, maybe $d=2$ and $G$ is the cyclic group of order $n$ acting by rotations. This seems to be the example you started out thinking of. Note that, when $n=2$, this representation is not irreducible, which gives you darij's counterexample.


Then I claim that $\sum a\_i a\_i^T = (n/d) \mathrm{Id}$.


Proof: $\sum a\_i a\_i^T$ is the matrix of a $G$ invariant bilinear form. Since $\mathbb{R}^d$ is irreducible, the space of such forms is one dimensional and $\sum a\_i a\_i^T = r \mathrm{Id}$ for some $r$. Taking traces of both sides, we get $n = rd$, so $r=n/d$.




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Here is this answer written in the most elementary way I can: Let's first look at the case you started with, where the vector are evenly spaced at angle $\theta$. Let $g(\theta)$ be the matrix $\left( \begin{smallmatrix} \cos \theta & \sin \theta \\ - \sin \theta & \cos \theta \end{smallmatrix} \right)$. Then we have
$$g(\theta) \left( \sum a\_i a\_i^T \right) g(-\theta) = \sum \left( g(\theta) a\_i \right) \left( g(\theta) a\_i \right)^T = \sum a\_i a\_i^T$$
where the second equality is because multiplying by $g(\theta)$ permutes the $a\_i$'s. So the sum $Q:=\sum a\_i a\_i^T$ obeys $g(\theta) Q g(-\theta) = Q$. This is a collection of four linear equations in the entries of $Q$. (Computationally, you may find it easier to work with the equivalent $g(\theta) Q = Q g (\theta)$.) As long as $\theta$ is not a multiple of $\pi$, the space of solutions is $1$-dimensional, spanned by the identity matrix. So $Q=r \mathrm{Id}$ for some $r$. Taking traces of both sides, $r=n/2$.


Now, what to do in higher dimensions? Are the vertices of a cube regularly spaced? What about a dodecahedron? The answer is yes, and group representation theory is the correct vocabulary to explain why. Let $g\_1$, $g\_2$, ..., $g\_k$ be a finite collection of orthogonal matrices permuting the $a\_i$ amongst themselves. So, the rotational symmetries of the cube or the dodecahedron or, back in your original example, the matrix $g(\theta)$. 


Set $Q = \sum a\_i a\_i^T$. Just like before, we deduce that $g\_1 Q = Q g\_1$, $g\_2 Q = Q g\_2$, ..., $g\_k Q = Q g\_k$. This is a bunch of linear equations for $Q$. If the space of solutions to these equations is one-dimensional, we win!


We now come to a nontrivial theorem: The space of solutions to these equations is one dimensional if and only if there is no linear subspace $V$ of $\mathbb{R}^d$ (other than $(0)$ and $\mathbb{R}^d$ itself) such that $g\_i V = V$ for each $g\_i$. For the cube, the dodecahedron, the original evenly spaced points, and many harder examples, this is easily checked. For the case darij raised, of two points evenly spaced at distance $\pi$, this condition fails: every line through the origin is taken to itself under $g(\pi)$. 


Some vocabulary: It is usual to work not just with the $g\_i$, but with all of their products. (For example, working with not just $g(\theta)$ but $g(2 \theta)$, $g(3 \theta)$, etcetera.) Let $G$ be the set of all of these products. Since they permute a finite set of vectors, $G$ cannot be infinite. (Experienced mathematicians will see that I am glossing over something here, please let me do so.) For example, in our two dimensional example, $\theta/\pi$ cannot be irrational. So $G$ is finite, and is what is called a **finite group**. Every element of $G$ is a symmetry of $\mathbb{R}^d$, so we say that $G$ **acts on $\mathbb{R}^d$** and those actions are by linear maps so we say that $G$ **acts linearly**. In this setting of a group acting linearly, we say that $\mathbb{R}^d$ is a **representation** of $G$. The condition that there be no subspace of $\mathbb{R}^d$ which is taken to itself by $G$ is usually stated using the technical term: **$\mathbb{R}^d$ is an irreducible representation**.


A personal note: If you aren't comfortable switching between matrices and bilinear forms, you don't know the vocabulary of groups and group actions, and you have never seen any group representation theory, you might fit in better at math.stackexchange.com than here. That said, nice problem! I may save it for when I teach representation theory.


