# Eigenvectors with at least one positive component

1. Oct 17, 2012

### 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.

2. Oct 18, 2012

### Erland

Re: Eigenvectors with atleast one positive component

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.

3. Oct 18, 2012

### 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.

4. Oct 19, 2012

### 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.

5. Oct 19, 2012

### 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.

6. Oct 19, 2012

### 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!

Last edited: Oct 19, 2012