Consider $n + 2$ vectors $v_1, v_2, v_3, \ldots, v_{n+2}$ in $\mathbb{R}^{n+2}$ with such property. For all $i, j$ the standart inner product $(v_j, v_i) < 0$. Denote by $v^{\bot}$ a subspace of $\mathbb{R}^{n+2}$ which is orthogonal to $v_1$ and denote by $v_j^{\bot}$ and orthogonal projection of $v_j$ onto $v^{\bot}$. How to prove that for all $i, j$ true that $(v_j^{\bot}, v_i^{\bot}) < 0$?
$n + 2$ vectors in $\mathbb{R}^{n}$ such that all their pairwise inner products are negative
linear algebra
Best Answer
Consider the explicit formula for the projected vectors
$$v_{j}^{\perp}=v_j-\frac{v_1\cdot v_j}{||v_1||^2}v_1~~~,~~~ j\in\{2,...,n+2\}$$
Then under the assumption that all inner products are negative you can show
$$v_i^{\perp}\cdot v_j^{\perp}=v_i\cdot v_j-\frac{(v_1\cdot v_i)(v_1\cdot v_j)}{||v_1||^2}<0$$
is negative as well.