How can I construct an orthonormal matrix in which the variance over the columns is minimal?
To be more formal, I would like to find a matrix $\mathbf{A} = [\mathbf{a}_1, \mathbf{a}_2, \ldots, \mathbf{a}_N], \mathbf{a}_i \in \mathbb{R}^N, \langle \mathbf{a}_i, \mathbf{a}_j \rangle = \delta_{ij}\ \forall\ i,j$, such that $\sum_{i=1}^N \sigma_i^2$ is minimal. $\sigma_i^2 = \sum_{j=1}^N (a_{ij} - \mu_i)^2$ is the variance of vector $\mathbf{a}_i$ and $\mu_i = \sum_{j=1}^N a_{ij}$ the mean.
Any ideas?