Edit: I needed to review this question as suggested.

Suppose there are $ N $ Realizations of the Gaussian process denoted as vectors $ mathbf {z} _ {j} in mathbb {R} ^ {n} $ for $ j = 1, ldots, N $. Leave $ and $ be a random variable such that $ y = sum_ {j = 1} ^ {N} ( mathbf {B} mathbf {z} _ {j}) (i) $

where $ mathbf {B} $ It is a unitary matrix. What is the variance of $ y2?

Explanation: Boldface represents the vector or matrix. $ ( mathbf {B} mathbf {x}) (i) $ represents the $ i $-th vector entry $ mathbf {B} mathbf {x} $.