Suppose a matrix in $\mathbb{R}^n$ has $n$ different real eigenvalues. There are now $n$ eigenvectors that are unique up to a scaling factor. By requiring them to be of unit length, they are unique up to a sign.
Now, I can try to define the first component of the vector to be positive in order to resolve the sign issue. However, not only do I run into problems when said component is zero, it also leads to discontinuities: When the eigenvector is changed by a small amount, the representation can jump to a different value.
Is there a unique representation that resolves this issue? The representation does not necessarily have to be a valid eigenvector itself.
All I can think of is to use the matrix $\frac{vv^\top}{\|v\|^2}$ to uniquely represent eigenvector $v$, but that isn't very compact as the number of components is squared.
In the special case of $\mathbb{R}^2$ I suppose I can obtain the angle of the vector, and construct a new vector with exactly twice the angle and unit length. Can this be extended to arbitrary dimensions.