Adaptive Eigendecomposition of Covariance Matrices

Hello,

I am facing the following problem and was wondering whether anybody
would be able to help.

In my application, a covariance matrix C gets updated over time and I
observe a sequence of matrices C_1, C_2, ... C_t. At each time point,
I need to do a eigen-decomposition of this matrix, but I do not have
access to the observation vector, only the matrix C.

Is there any efficient way to update the eigen-decomposition at each
time step by only using C?

I am aware of some RLS-like methods for updating the eigen-
decomposition of C_t adaptively but they all require access to the
data vector.

The reason why I don't have access to the data vector is because C_t
is actually computed in the following way:

C_t = rho A_t + (1-rho) B_t with rho in [0,1]

where A_t and B_t are two covariance matrices that get updated on-line
in the usual way

A_t = mu A_t-1 + (1-mu) xx'
B_t = mu B_t-1 + (1-mu) x'y

where mu is a smoothing parameter in [0,1]. I do have access to data
vectors x and y.

Many thanks,

John