complex numbersmatricesnormed-spaces
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.
Almost four years and no answers? I can't give you a complete answer, but I guess under the theory of better late than never I will try to address as many as your points as I can. Though it's possible that by now you have gotten a better answer through another source.
Does anyone have any good resources on how complex number systems work?
Yeah. Go to Wikipedia and scroll all the way to the bottom, to find the external links. The Gilbert paper looks very good and transparent, and it's available as a PDF. Your local university library might have a copy of Knuth's monumental 3-volume book.
there are a couple of things I don't quite understand (why there are 4 digits, for example).
This is a very interesting question in its own right. I think that first you need to understand "negabases" (numeral systems with a negative number as the base), but before that, I will review some very basic basics of positive integer bases.
You already know, at least on an intuitive level, the basic principle of power series numeral systems with a positive integer as the base. Given a base $b$ and a positive integer $n$ represented by the string of $\mathcal{L}$ digits $\{d_{\mathcal{L} - 1}, d_{\mathcal{L} - 2}, \ldots d_1, d_0 \}$, the string of digits is essentially a shorthand for $$n = \sum_{i = 0}^{\mathcal{L} - 1} d_ib^i$$. Of course this basic principle also works when $b$ is almost anything other than a positive integer greater than 1.
I could be wrong on this next point, but I think that there are only three numbers that are completely unworkable as bases for a power series positional numeral system, namely $-1$, 0 and 1. I think that for a base $b$ to be "workable" for this purpose, the following needs to be true: if $\alpha$ and $\beta$ are distinct integers drawn from $\mathbb{Z}$, then $b^\alpha$ and $b^\beta$ are distinct numbers.
With $b = -1$, $\alpha$ and $\beta$ can be distinct but if they're of the same parity, then $b^\alpha = b^\beta$, and with $b = 1$ they can have different parities. It should be obvious that 0 is completely useless as a base. Other than that, I guess any number whatsoever can work as a base. Of course some bases have more practical value than others. For example, quite a few people think we'd all be better off replacing 10 with 12 as our main number base, but I've never heard anyone say the same for $\sqrt[3]{1729}$.
With complete confidence I can tell you that there are only two digits you must absolutely have in every power series positional numeral system: 0 and 1. You have to have 0 to signify the absence of a specific power of the base. Whether you need any more digits depends on the choice of base. Given an integer $b > 2$, you also need a digit for $b - 1$; in fact you need to be able to represent the integers from 0 to $b - 1$ with a single digit.
It is this need for digits from 2 to $b - 1$ when $b > 2$ that leads to some frustrated expectations when the base is not a positive integer. For example, for $b = 4$, you need digits for 0, 1, 2, 3. You don't need a digit for 4 because that's represented as 10. What digits do you need for "negaquartal," with $b = -4$? Digits for $-1$, $-2$, $-3$? Actually, no (though I suppose that if I really wanted to, I could try to work out a system using such digits; I have not wanted to).
In a philosophical sense, systems with positive bases are incapable of representing negative numbers. Sure, we have the negation operator, but in a sense that is actually a shorthand for "$0 -$." Which is to say that, e.g., $-7$ doesn't represent negative seven, that it actually represents $0 - 7$. It is this perceived shortcoming that negabases address: they can represent negative numbers without the need for a negation operator.
For negaquartal, you need digits for 0, 1, 2, 3. The numbers $-1$, $-2$, $-3$ are not single-digit numbers in this system. But you can't use 10 for 4 because now that means $-4$. So positive 4 is represented as 130, and in general, for $\alpha$ odd, the negaquartal representation of $4^\alpha$ is 13 followed by $\alpha$ 0s; this is tantamount to $4^{\alpha + 1} - 3(4^\alpha)$. The negaquartal representation of $4^\alpha$ with $\alpha$ even is the same as with positive quartal. As for $-3$, that's 11, which means $-4 + 1$.
To represent complex numbers, both positive bases and negative bases have to separate the real part from the imaginary part. That's where complex number bases come in. Quater-imaginary, as you know, has $b = 2i$. As you may have already realied, $2i$ is the principal square root of $-4$. So the way it works out is that to obtain the quater-imaginary representation of a purely real integer, be it negative or positive, you can just take the negaquartal representation and riffle in some zeroes. Thus, $-3$ is 11 in negaquartal and 101 in quater-imaginary. I know I've been longwinded on this one without fully answering your question, but if you've read this far and you understand why quater-imaginary uses 4 digits, I'm happy.
The matrix $p$-norm is the operator norm: $$ \|A\|_p=\max_{x\neq 0}\frac{\|Ax\|_p}{\|x\|_p}. \tag{1} $$ The fact that $$\tag{2} \|A\|_2=\rho(A^*A)\quad\text{and}\quad\|A\|_{\infty}=\max_i\sum_j|a_{ij}| $$ is just the consequence of (1). The $2$-norm and $\infty$-norm satisfy $$\tag{3} \|x\|_{\infty}\leq\|x\|_2\leq\sqrt{n}\|x\|_{\infty} $$ for any $n$-vector $x$ and the fact that $\|A\|_2\leq\sqrt{n}\|A\|_{\infty}$ follows by plugging (3) to (1): $$ \|A\|_2=\max_{x\neq 0}\frac{\|Ax\|_2}{\|x\|_2}\leq \max_{x\neq 0}\frac{\sqrt{n}\|Ax\|_\infty}{\|x\|_2} \leq \max_{x\neq 0}\frac{\sqrt{n}\|Ax\|_\infty}{\|x\|_{\infty}}=\sqrt{n}\|A\|_{\infty}. $$
Assuming that you know nothing about (1) and take (2) as the definitions (such an approach in a class is a bit odd because it makes showing a lot of things quite complicated), you can proceed as follows:
Best Answer
$\def\RR{\mathfrak{R}}\def\II{\mathfrak{I}}\def\ii{\iota}$ 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: $$ \begin{align} \|Ax-b\|_2^2 &= \|(\RR A + \ii\II A)(\RR x + \ii\II x)-(\RR b + \ii\II b)\|_2^2 \\ &= \|\RR A \RR x - \II A \II x -\RR b + \ii( \RR A \II x + \II A \RR x - \II b )\|_2^2 \\ &= \|\RR A \RR x - \II A \II x -\RR b\|_2^2 + \|\RR A \II x + \II A \RR x - \II b \|_2^2. \end{align} $$