# multiplication of a binary vector by an orthonormal matrix

Suppose we have a orthonormal matrix $V \in \mathcal{R}^{n \times n}$ (i.e. $V V^T = I$). The columns of the $V$ matrix are othogonal unit vectors.

Build the matrix $V_k$ composed of the first $k < m$ columns of $V$.

Then take a vector $x \in \lbrace 0, 1\rbrace^n$ and multiply: $$y = xV_k V_k^T$$

Is it possible that for some $i \in \lbrace 1, 2, \dots, n \rbrace$ we have: $$y_i > 1$$ ?

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Yes, it is possible that some entries of $y$ exceed 1. Let $u=\frac1{\sqrt{3}}(1,1,1)^T,\ v=\frac12((1,0,0)^T-u)$ and $V = I - 2\frac{vv^\top}{\|v\|^2}$. So $V$ is a real orthogonal matrix with $u$ as its first column. Now, $$(1,1,0)V_2V_2^T \approx (1.1220, 0.8333, 0.0447).$$