The Norm
//Nine Dragon Tampan.// Paminggir people. Sumatra circa 1900, 41 x 43 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
Nine Dragon Tampan. Paminggir people. Sumatra circa 1900, 41 x 43 cm. From the library of Darwin Sjamsudin, Jakarta. Photograph by D Dunlop.
Consider an ordered-set of three arbitrary numbers $a_{1}$, $a_{2}$ and $a_{3}$ that collectively constitute some algebraic vectorXlink.png that is generically written as

$\overline{a} = \left( \, a_{1}, a_{2}, a_{3} \right)$

These three numbers can be compressed into a single number called the norm of $\overline{a}$. We usually mark the norm of a vector using symbols like $\left\| \, \overline{a} \, \right\|$ or often just $a$. The norm depends on nine constants that are collectively called a metric. They are written using $k$ with a subscript, for example $k_{12}$. The numerical values for these constants are established by the context of a calculation, they are often implicit. Definition: the norm of $\overline{a}$ is

$\parallel \overline{a} \parallel \; \equiv \sqrt{ \sum_{j=1}^{3} \sum_{k=1}^{3} k_{j k} a_{j} a_{k} \ }$

or more explicitly

$\left\| \, \overline{a} \, \right\| = \sqrt{ \vphantom{\sum^{2} } \ k_{11} a_{1}^{2} + k_{22} a_{2}^{2} + k_{33} a_{3}^{2} + ( k_{12} + k_{21}) a_{1} a_{2} + ( k_{13}+ k_{31} )a_{1} a_{3} + ( k_{23}+ k_{32} )a_{2} a_{3} \ \ }$


Summary
Noun Definition
Norm $\parallel \overline{a} \parallel \; \equiv \sqrt{ \sum_{j=1}^{3} \sum_{k=1}^{3} k_{j k} a_{j} a_{k} \ }$ 5-17
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