2-Norm of a complex matrix equation

I am having trouble understanding the following excerpt from a math text I'm working through:

My question specifically is how line 2 came about in the expansion. How do the real and imaginary parts expand like that? Also I can't figure out from the context what the cursive "I" means.

For reference, I understand how to expand the 2-norm squared of $||x + iy||^2 = x^2 + y^2$, but I can't seem to understand how it's working in this example.

Thanks.

• What is the "cursive 'I'"? I mean, what are you talking about? – Git Gud May 27 '14 at 16:56
• I don't really know myself, I was hoping it would be obvious to someone from context =/ – JDS May 27 '14 at 17:00
• You misunderstood me. I don't know what you're talking about. Do you perhaps mean $\Im$? – Git Gud May 27 '14 at 17:01
• Anyway, $\Im$ denotes the imaginary part. If you don't know this,then I'm really confused because you were able to note that the real and imaginary parts came into play here ("How do the real and imaginary parts expand like that?"). And if you know this, then I ask again: what are you talking about? – Git Gud May 27 '14 at 22:43

$\def\RR{\mathfrak{R}}\def\II{\mathfrak{I}}\def\ii{\iota}$ You might notice that $(\alpha+\ii\beta)(\gamma+\ii\delta)=\alpha\gamma-\beta\delta+\ii(\beta\gamma+\alpha\delta)$, where $\alpha,\ldots,\delta\in\mathbb{R}$ and $\ii$ is the imaginary unit. Similar stuff holds, e.g., for matrix-vector products. So: \begin{align} \|Ax-b\|_2^2 &= \|(\RR A + \ii\II A)(\RR x + \ii\II x)-(\RR b + \ii\II b)\|_2^2 \\ &= \|\RR A \RR x - \II A \II x -\RR b + \ii( \RR A \II x + \II A \RR x - \II b )\|_2^2 \\ &= \|\RR A \RR x - \II A \II x -\RR b\|_2^2 + \|\RR A \II x + \II A \RR x - \II b \|_2^2. \end{align}