Eigenvectors with at least one positive component

[rkrsnan]

I am wondering if there is a systematic way to fix the phase of complex eigenvectors. For example e^{i \theta}(1,\omega,\omega^2) where e^{i \theta} is an arbitrary phase and \omega and \omega^2 are the cube roots of unity, is an eigenvector of the cyclic matrix \left(\begin{matrix}0& 1&0\\0&0&1\\1&0&0\end{matrix}\right). But I feel that (1,\omega,\omega^2) and equivalently (\omega,\omega^2,1) and (\omega^2,1,\omega) are somehow special because one of the components is real and positive. Is there some special name for such a choice of phase? Any reference will be greatly appreciated.

 

[Erland]

I am not sure what you really want, but any eigenvector has at least one nonzero component, call it a. Dividing the vector with a gives an eigenvector such that (at least) one of its components is 1, which is real and positive.

 

[AlephZero]

The phase of an eigenvector is arbitrary, whether it is real or complex.

If you are finding the solution to some problem as a linear combination of eigenvectors $\sum_i a_i \phi_i$, you could say that each "component' $a_k \phi_k$ has a definite phase (and magnitude) because the numerical value of $a_k$ depends on the arbitrary choice for the phase of the corresponding $\phi_k$.

In some practical situations you might choose to fix the amplitude and phase of an eigenvector in an arbitrary way (e.g. by making the biggest component of the vector = +1) but doing that doesn't have much mathematical significance.

 

[rkrsnan]

Thanks guys for the reply. I am aware of the things you pointed out. As Erland said I should have been more specific on what I want. I am studying the representations of discrete groups for application in physics. Eigenvectors of group elements are significant in this context (for example the matrix that I wrote in the earlier post represents a rotation by 2pi/3 about the axis (1,1,1) and the axis (1,1,1) is an eigenvector). The problem of phase arises when I starts looking at complex eigenvectors for example (1, \omega,\omega^2). All I am saying is that a phase convention with one of the components real and positive is "appealing" to me (especially in the physics context) and so I want to know if such a choice is somehow special for a mathematian.

 

[rkrsnan]

Ohh, I now notice something. Since my representation matrices are unitary, their eigenvalues are always in the form e^{i \theta}. For the matrix that I wrote in the first post, the eigenvalue (corresponding to the eigenvector (1,\omega, \omega^2)) is e^{i 2\pi/3}. So basically the "appealing" phase convention has all its components having phases just a multiple of the eigenvalue.

So let me make a hypothesis. Please let me know it is true.

Let e^{i \theta} be an eigenvalue for a special unitary matrix, U. We will always be able to choose a phase convention for the corresponding eigenvector so that all its components have a phase e^{i n \theta} where n is an integer.

 

[rkrsnan]

Ok, I was wrong, I tried with random unitary matrices and it didn' t work. So let me modify my hypothesis.

Let U be a discrete unitary matrix, with U^n = I, where n is an integer. Its eigenvalues will be of the form e^{i m\frac {2\pi} {n}} where m is an integer. There exists a basis and a phase convention in which all the components of the eigenvectors are of the form | a_x | e^{i k_x\frac {2\pi}{n}} where k_x are integers.

Please let me know if the above statement is true. Thanks very much!