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.



Linear combination of orthogonal projection matrices
If you make the change Mikael suggests, you are basically looking at tight frames. $c\_i^{1/2} v\_i$ should be the image under an orthogonal projection of an orthonormal basis from some $n$ dimensional superspace.


