principal eigenvectors of an unknown matrix

Question

Do you have any idea about how we can find the principle eigenvectors of an unknown matrix ${H}$. The only information that we have is that $H$ has only a few (up to 3) dominant eigen modes regardless of its dimension. The power method is one solution, but the complexity is huge, especially because it requires many iterations to find the dominant eigenvectors.

Answer 1

The elements of $H$ are unknown in general. If you are familiar with channel estimation procedure in wireless communications, $H$ is channel matrix, and every element $(i,j)$ represents the channel gain between antenna element $i$ of transmitter and antenna element $j$ of the receiver. Assume that we cannot afford estimation of every element of $H$. Still, we can estimate (means that we have) the second order statistics, such as covariance matrix of the channel $E\{HH^T\}$. Another useful information is that we know the channel has only up to 3 dominant eigenmodes.

Now, I am interested to find dominant eigenvectors, correspond to dominant eigenmodes, of the channel matrix $H$.