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I have a set of unknown vectors $v_1, v_2,\ldots,v_n$, but I know what are supposed to be all inner products $p_{ij}=v_i\cdot v_j$ between them.

Is there is kind of inequality of equality that tell me if the inner product values are valid given some dimension of the vector space?

EDIT: To be more concrete, in my case I have pairs of vectors $a_i$, $b_i$ and I know $$ \begin{align*} a_i^2&=b_i^2=1\\ a_i\cdot b_i&=0\\ a_i\cdot a_j&=0.5\qquad (i\neq j)\\ a_i\cdot b_j&=0.5\qquad (i\neq j)\\ b_i\cdot b_j&=0.5\qquad (i\neq j) \end{align*} $$ I wonder about the relation of the number of those pairs and the dimension of the vector space.

Gere
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  • Start with the simple case of two vectors. What can you infer? – Phil Freedenberg Apr 24 '22 at 16:58
  • without more structure the best you can do is look at the signature of the Gram matrix. If there is a lot more structure, e.g. this https://math.stackexchange.com/questions/4329566/let-v-be-an-euclidian-space-of-dimension-n-and-let-v-1-v-m-in-v-s-t/ may be of interest – user8675309 Apr 24 '22 at 19:58
  • There is more structure. There are pairs which are orthonormal and across pairs the inner product is always 0.5. What is the relation between the number of pairs and the dimension? @user8675309 – Gere Apr 24 '22 at 20:08

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