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Suppose I have a discrete sequence (or would it be a set?) of numbers labeled by two integer indices, $\{a_{ik}|i,k\in 1,\ldots,n\}$. Also, I know that these numbers satisfy the properties

$$a_{ij} + a_{jk} = a_{ik}$$

and

$$a_{ik} = -a_{ki}$$

Are these conditions sufficient for me to conclude that there exists a sequence $\{\lambda_b|b\in 1,\ldots,n\}$ which satisfies

$$a_{ik} = \lambda_i - \lambda_k$$

for all $i,k$? If so, are both conditions necessary? If not, what other conditions would I require?


I really have no idea what tags to put on this, so I invite someone to retag it appropriately and remove this sentence.

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1 Answer 1

up vote 2 down vote accepted

Let $\lambda_1$ be arbitrary and define $\lambda_i = \lambda_1 + a_{1i}$. Now you have $a_{1i} = \lambda_i - \lambda_1$, and so $$a_{ij} = a_{1j} - a_{1i} = \lambda_j - \lambda_i.$$ Conversely, if there are numbers $\lambda_i$ such that $a_{ij} = \lambda_i - \lambda_j$ then clearly everything you wrote is true. So your conditions are both necessary and sufficient.

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I think some of your subscripts are in the opposite order they should be to match the way I wrote the question, e.g. $a_{1i} = \lambda_1 - \lambda_i$, not the other way around. But anyway, thanks :) I had a feeling it'd be something like that. –  David Z Feb 16 '11 at 20:29

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