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While reading through some engineering related journals, I've come across the following notation:

$$\|p − Xw\|^2_2$$

$p$ and $w$ are vectors, while $X$ is a matrix. I understand that $\|p − Xw\|$ is finding the norm of the vector, i.e. $\left\|{\boldsymbol {x}}\right\|:={\sqrt {x_1^2+\cdots +x_n^2}}$. But what does two 2s mean at the subscript and superscript? I've seen them in multiple places, so I assume that it is a common notation, yet I could not find the exact definition online (it's hard to search when I don't know the terminology nor know how to type such mathematical function in google).

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  • $\begingroup$ The common notation is that the subscript 2 means that it's the 2-norm (as opposed to the 3-norm, 1-norm, or whatever), while the superscript is just an exponent 2. $\endgroup$
    – Nathan H.
    Mar 23, 2017 at 4:43
  • $\begingroup$ One can define $\| x \|_p = \sqrt[p]{\sum_{i=1}^{n} |x_i|^p}$ for $p > 0$, which generalizes the concept of a 2-norm. The subscript is denoting $p=2$, and the superscript is just squaring. $\endgroup$ Mar 23, 2017 at 4:44

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A norm is actually something much more general than simply the expression you gave. It is simply a function that satisfies certain properties. However, it turns out that the $2$-norm is exactly the norm you are used to.

In general, the $p$-norm of a vector is given by $$\|\boldsymbol {x}\|_p = \big(|x_1|^p + |x_2|^p + ... + |x_n|^p\big)^{1/p}.$$

You will see that by plugging in $p = 2$ you get the norm you are used to.

The superscript simply refers to ordinary squaring, hence $$\|\boldsymbol{x}\|_2^2 = |x_1|^2 + |x_2|^2 + ... + |x_n|^2.$$

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    $\begingroup$ Those $|\cdot |$'s are just absolute values right? $\endgroup$ Oct 21, 2021 at 7:32
  • $\begingroup$ @RounakSarkar Yes, just regular ol' absolute value. $\endgroup$
    – Eff
    Oct 26, 2021 at 11:50

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