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My book states the following for a generalisation on $\mathbb{R}^k$: $\displaystyle\sum_{i,j=1}^{k} a_{ij}x_ix_j$

Is this the same notation as $\displaystyle\sum_{i\leq j=1}^{k} a_{ij}x_ix_j$ or is it a mistake in the book?

Aqancake
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1 Answers1

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The first expression you gave is symmetric, in that it is unchanged when you replace $a_{ij}$ by $a_{ji}$. If you do so, add it to the original, and divide it by $2$, you have the same expression, but with $a_{ij}$ replaced by $\frac12(a_{ij}+a_{ji})$. This means that you can assume, without loss of generality, that $a_{ij}=a_{ji}$ in the first form.

But then, if you put $b_{ii}=a_{ii}$ and $b_{ij}=2a_{ij}$ when $i<j$, you get $$\sum_{i,j}a_{ij}x_ix_j=\sum_{i<j}b_{ij}x_ix_j.$$

So the two forms are equivalent, at the cost of some tweaking of the exact coefficients.

Or more briefly, just let $b_{ii}=a_{ii}$ and $b_{ij}=a_{ij}+a_{ji}$ when $i\ne j$, and skip the discussion of symmetry.

Harald Hanche-Olsen
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