Hello!

Let $v_i\in R^d$ $(i=1,...,n,n>d)$ be unit-length vectors ($v_i^Tv_i=1$). Then $v_iv_i^T$ is an *orthogonal projection matrix*, which has many elegant properties. Now consider a linear combination of these orthogonal matrices
$$A=\sum_{i=1}^n c_i v_i v_i^T$$
where $c_i$ are positive scalars and $\sum_{i=1}^n c_i=1$. **My question is**: what $v_i$ can make the linear combination of these orthogonal matrices be an identity matrix? That is: how to find $v_i$ such that
$$\sum_{i=1}^n c_i v_i v_i^T=\frac{1}{d}I$$

Thanks.