Suppose vectors $a_p\in \mathbb{R}^n$, $p\in [n]$ form an orthonormal basis. For a given unit vector $b\in \mathbb{R}^n$, the summation $\sum_{p\in[n]} b^Ta_p$ is at its maximum ($=\sqrt{n}$) if $b^Ta_p = \frac{\sqrt{n}}{n}$ for all $p\in[n]$.

Given unit vectors $b, c\in \mathbb{R}^n$, my question is what is a tight upper bound for the following summation $s=\sum_{p\in[n]} b^Ta_p c^Ta_p$ assuming $b^T c = \alpha$.

(for simplicity, we can assume both unit vectors $b$ and $c$ are in the positive orthant.)

  • $\begingroup$ $$\sum_p b^Ta_pc^Ta_p = b^T\sum_pa_p(c^Ta_p) = b^Tc = \alpha$$ $\endgroup$ Aug 9, 2022 at 21:29
  • $\begingroup$ Thank you so much. $\endgroup$ Aug 9, 2022 at 21:34


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