The rows of this unitary matrix form an orthonormal set but the columns don't. Why?

The rows of this ( {{1/3 - 2/3i,2/3i},{-2/3i,-1/3-2/3i}} - Wolfram|Alpha ) unitary matrix form an orthonormal set but, the columns of A do not form an orthonormal set despite this ( https://en.wikipedia.org/wiki/Unitar...ent_conditions ) Wikipedia article saying that they should.Could someone please help me understand why the columns of A do not form an orthonormal set for this unitary matrix?

Re: The rows of this unitary matrix form an orthonormal set but the columns don't. Wh

Hmm...let's see:

Let's take the inner product of the first two columns:

$\displaystyle \left(\frac{1}{3} - \frac{2}{3}i\right)\left(\overline{\frac{2}{3}i} \right) + \left(-\frac{2}{3}i \right)\left(\overline{-\frac{1}{3} - \frac{2}{3}i} \right)$

$\displaystyle =\left(\frac{1}{3} - \frac{2}{3}i\right)\left(-\frac{2}{3}i \right) + \left(-\frac{2}{3}i \right)\left(-\frac{1}{3} + \frac{2}{3}i \right)$

$\displaystyle = -\frac{4}{9} - \frac{2}{9}i + \frac{4}{9} + \frac{2}{9}i = 0$.

So it appears the columns are orthogonal.

Taking norms, for the first column we have:

$\displaystyle \sqrt{\left(\frac{1}{3} - \frac{2}{3}i \right)\left(\frac{1}{3} + \frac{2}{3}i \right) + \left(-\frac{2}{3}i \right)\left(\frac{2}{3}i \right)}$

$\displaystyle = \sqrt{\frac{1}{9} + \frac{4}{9} + \frac{4}{9}} = \sqrt{1} = 1$.

I leave it to you to show that the second column is also a unit vector. I don't see the problem.

Re: The rows of this unitary matrix form an orthonormal set but the columns don't. Wh

I had just figured out that I was doing the dot products incorrectly and was going to mention it ... sorry that you had to type that out but, thank you!