-1
$\begingroup$

I am looking for a general way to write an orthonormal basis in $\mathbb{C}^2$.

For example, one could write a general orthonormal basis in $\mathbb{R}^2$ as:

$V=a\vec{x}+b\vec{y}\;$ for $\;a,b\in\mathbb{R}$ and

$V^\perp=-b\vec{x}+a\vec{y}\;$ for $\;a,b\in\mathbb{R}$

Thus I would like something of the form:

$Q=a\vec{x}+b\vec{y}\;$ for $\;a,b\in\mathbb{C}$ and

$Q^\perp=c\vec{x}+d\vec{y}\;$ for $\;c,d\in\mathbb{C}$

such that $=0$ and $c=f(a,b)$ and $d=g(a,b)$

  • 0
    I don't see where you have written an orthonormal basis, or any kind of basis, for ${\bf R}^2$. A basis for ${\bf R}^2$ is a set of two elements of ${\bf R}^2$ (with certain special properties), and I don't see where you have named any vectors of ${\bf R}^2$. I don't know what you mean by $V$, and I don't think $V^{\perp}$ has anything to do with an orthonormal basis – an orthonormal basis is one in which the two elements forming the basis are unit vectors and are normal to each other. Maybe if you rethink your understanding of orthonormal basis for ${\bf R}^2$, you'll understand ${\bf C}^2$. – 2017-01-26
  • 0
    As I understand, $V$ and $V^\perp$ are orthogonal to one another - I did forget to mention that the sum of magnitude of the coefficients for each vector is constrained to 1. – 2017-01-26
  • 0
    Any thoughts about my answer? – 2017-01-28
  • 0
    Are you still here? – 2017-01-30

1 Answers 1

0

Given any vector $v=(a,b)$ in ${\bf R}^2$, its length $r$ is given by the formula $r=\sqrt{a^2+b^2}$. If $v$ is not the zero vector, then $(a/r,b/r)$ is a unit vector in the direction of $v$. Then $\{\,(a/r,b/r),(b/r,-a/r)\,\}$ is an orthonormal basis for ${\bf R}^2$ (with respect to the usual inner product). Moreover, every orthonormal basis for ${\bf R}^2$ is of that form, or of the form $\{\,(a/r,b/r),(-b/r,a/r)\,\}$.

Given any vector $v=(a,b)$ in ${\bf C}^2$, its length $r$ is given by the formula $r=\sqrt{|a|^2+|b|^2}$. If $v$ is not the zero vector, then $(a/r,b/r)$ is a unit vector in the direction of $v$. Then $\{\,(a/r,b/r),(\bar b/r,-\bar a/r)\,\}$ is an orthonormal basis for ${\bf C}^2$ (with respect to the usual inner product). Moreover, every orthonormal basis for ${\bf C}^2$ is of that form, or of the form $\{\,(a/r,b/r),(-\bar b/r,\bar a/r)\,\}$.

Here, $\bar w$ denotes the complex conjugate of the complex number $w$.