# Normalising Imaginary Eigenvector

1. Nov 3, 2015

### BOAS

Hello,

whilst solving a system of coupled differential equations I came across an eigen vector of $\vec{e_{1}} = (^{1}_{i})$.

Assuming that this is a correct eigenvector, how do I normalise it? I want to say that $\vec{e_{1}} = \frac{1}{\sqrt{2}} (^{1}_{i})$ but if I sum $1^{2} + i^{2}$ I get zero.

It seems sensible to me that the vector's length is root two, but how do I justify this, if at all?

Thank you.

2. Nov 3, 2015

### Orodruin

Staff Emeritus
In a complex vector space, you have to introduce an inner product which satisfies $\langle x, y\rangle = \overline{\langle y,x\rangle}$.

3. Nov 3, 2015

### davidmoore63@y

The squared length of a complex vector v is defined by v.v(bar) where v(bar) is the complex conjugate, i believe. That will give you sqrt2

4. Nov 4, 2015

### BOAS

Ah of course. A complex number is essentially a vector.

Thank you.

5. Nov 4, 2015

### Orodruin

Staff Emeritus
I agree if you take away the "essentially". It is an element in a complex one-dimensional vector space. :)