1.This is a question from paper "Fast Generalized Eigenvector Tracking Based on the Power Method".
2.The author wrote "We generate two zero-mean Gaussian random vectors ,which have correlation matrices A and B whose eigenvalues are exponentially distributed".
3.But how to generate a zero-mean Gaussian random vector ,which have correlation matrices whose eigenvalues are exponentially distributed ,this confused me almost a week.
4.It seems that we could only use randn in MATLAB to generate random vector, so the problem is how to make sure correlation matrices whose eigenvalues exponentially distributed at the same time?
Let S be a positive definite matrix. Therefore S has a Cholesky decomposition L.L' = S where L is a lower-triangular matrix and ' denotes the matrix transpose and . denotes matrix multiplication. Let x be drawn from a Gaussian distribution with mean zero and covariance equal to the identity matrix. Then y = L.x has a Gaussian distribution with mean zero and covariance S.
So if you can find suitable covariance matrices A and B, you can use their Cholesky decompositions to generate samples. Now about constructing a matrix which has eigenvalues following a given distribution. My advice is to start with a list of samples from an exponential distribution; these will be your eigenvalues. Let E = a matrix with the exponential samples on the diagonal and zeros otherwise. Let U be any unitary matrix (i.e. columns are orthogonal and norm of each column is 1). Then U.E.U' is a positive definite matrix with the specified eigenvalues.
U can be any unitary matrix. In particular U can be the identity matrix. That might make everything else simpler; you'll have to verify whether U = identity is workable for the problem you're working on.